271
Session 1
Finance 2
Financial Modeling and Econometrics
Philip W. Wirtz
The George Washington University
Clicker Check
According to the Arizona Daily Sun, Flagstaff Arizona police
to the Arizona Daily Sun Flagstaff Arizona police
had to calm residents when local drivers making their morning
commute were greeted by an electronic sign along a busy road
warning of
1. an escaped criminal on the loose
2. a rogue panda on a rampage
rogue panda on rampage
3. an escaped peacock from the zoo
4. killer bees heading into town
5. the closing of all Starbucks
2
NonNon-Response
Grid
Clicker Check
According to the Arizona Daily Sun, Flagstaff Arizona police
to the Arizona Daily Sun Flagstaff Arizona police
had to calm residents when local drivers making their morning
commute were greeted by an electronic sign along a busy road
warning of
0%
1. an escaped criminal on the loose
2. a rogue panda on a rampage
rogue panda on rampage
3. an escaped peacock from the zoo
4. killer bees heading into town
5. the closing of all Starbucks
1
2
3
4
3
5
NonNon-Response
The deed was apparently perpetrated by pranksters.
Grid
Source: http://azdailysun.com/news/local/police-nohttp://azdailysun
rogue-pandas-about/article_420be32f-7571-55079ce9-58b6f6ea8d4f.html
Administrivia
To receive full credit, assignments must have your correct GWID.
Contacting me on Friday is unlikely to yield a response: Faculty meetings and
Faculty Senate meetings are scheduled on Fridays.
I send out email to your GWU address on a very regular, and time-sensitive,
basis If necessary be sure to forward your gwu
basis. If necessary, be sure to forward your gwu.edu email.
email
If you havent completed the Student Information Record by the end of the day
today, your Assignment 2 grade will be reduced without recourse.
Whenever you write me about question involving the Gradebook
Whenever you write me about a question involving the Gradebook or a Master
Master
Key, please be sure to include your PIN in your correspondence with me.
4
5
Quiz Begins
Sold For
You wish to find the relationship between the Bluebook
value of a car (the independent variable) and what it sells
for at a local used car dealer (the dependent variable),
using five recently-sold cars as data. Based on these data,
what is the approximate value of the slope
what is the approximate value of the slope?
$30,000
$25,000
$20,000
$20,000
$15,000
$10,000
$5,000
$0
$0
$5,000 $10,000 $15,000 $20,000 $25,000 $30,000
Bluebook Value
0
1
2
3
4
5
6
7
8
9
60
Countdown
6
-3
-2.00
-1.00
-0.50
0.00
0.50
1.00
2.00
3.00
5
NonNon-Response Grid
7
Sold For
You wish to find the relationship between the Bluebook
value of a car (the independent variable) and what it sells
for at a local used car dealer (the dependent variable),
using five recently-sold cars as data. Based on these data,
what is the approximate value of the slope
what is the approximate value of the slope?
$30,000
$25,000
$20,000
$15,000
$10,000
$5,000
$0
$0
$5,000 $10,000 $15,000 $20,000 $25,000 $30,000
Bluebook Value
0
1
2
3
4
5
6
7
8
9
60
Countdown
-3
-2.00
-1.00
-0.50
0.00
0.50
1.00
2.00
3.00
5
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
8
Sold For
What is the approximate value of the intercept?
$30,000
$25,000
$20,000
$20,000
$15,000
$10,000
$5,000
$0
$0
$5,000 $10,000 $15,000 $20,000 $25,000 $30,000
Bluebook Value
0
1
2
3
4
5
6
7
8
9
-30,000
-22,500
-15,000
5,000
0
5,000
15,000
22,500
30,000
37,500
NonNon-Response Grid
15
Countdown
9
Sold For
What is the approximate value of the intercept?
$30,000
$25,000
$20,000
$15,000
$10,000
$5,000
$0
$0
$5,000 $10,000 $15,000 $20,000 $25,000 $30,000
Bluebook Value
0
1
2
3
4
5
6
7
8
9
-30,000
-22,500
-15,000
5,000
0
5,000
15,000
22,500
30,000
37,500
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
15
Countdown
10
Data
Please take 90 seconds to enter the data below into your computer or
calculator.
You will be making frequent reference to these data on the quiz, so please
record them carefully
record them carefully.
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
11
What is the value1 of SXX?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
209,720,920
209,720,921
209,720,922
209,720,923
209,720,924
209,720,925
209,720,926
209,720,927
209,720,928
209,720,929
1To
select the correct answer, identify that selection which is
closest to the exact value. For this question and all other
questions on the quiz, please use this rule to identify the
correct response
correct response.
NonNon-Response Grid
60
Countdown
12
What is the value1 of SXX?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
209,720,920
209,720,921
209,720,922
209,720,923
209,720,924
209,720,925
209,720,926
209,720,927
209,720,928
209,720,929
0
0
0
0
0
0
0
0
0
0
1To
select the correct answer, identify that selection which is
closest to the exact value. For this question and all other
questions on the quiz, please use this rule to identify the
correct response
correct response.
NonNon-Response Grid
60
Countdown
If we consider these 5 observations to be the population,
what is the standard deviation of the dependent variable?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
13
15,883
11,231
9,170
7,942
7,103
6,484
6,003
5,616
5,294
5,023
NonNon-Response Grid
60
Countdown
If we consider these 5 observations to be the population,
what is the standard deviation of the dependent variable?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
15,883
11,231
9,170
7,942
7,103
6,484
6,003
5,616
5,294
5,023
14
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
60
Countdown
What is the value of SXY?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
15
201,543,400
201,543,401
201,543,402
201,543,403
201,543,404
201,543,405
201,543,406
201,543,407
201,543,408
201,543,409
NonNon-Response Grid
120
Countdown
16
What is the value of SXY?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
201,543,400
201,543,401
201,543,402
201,543,403
201,543,404
201,543,405
201,543,406
201,543,407
201,543,408
201,543,409
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
120
Countdown
What is the value of the slope?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
17
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99
NonNon-Response Grid
45
Countdown
What is the value of the slope?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
.90
.91
.92
.93
.94
.95
.96
.97
.98
.99
18
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
45
Countdown
What is the value of the intercept?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
19
-1,820
-1,821
-1,822
-1,823
-1,824
-1,825
-1,826
-1,827
-1,828
-1,829
NonNon-Response Grid
45
Countdown
What is the value of the intercept?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
-1,820
-1,821
-1,822
-1,823
-1,824
-1,825
-1,826
-1,827
-1,828
-1,829
20
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
45
Countdown
21
What is the residual of Automobile Number 1?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
-1,600
-1,601
-1,602
-1,603
-1,604
-1,605
-1,606
-1,607
-1,608
-1,609
NonNon-Response Grid
30
Countdown
22
What is the residual of Automobile Number 1?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
-1,600
-1,601
-1,602
-1,603
-1,604
-1,605
-1,606
-1,607
-1,608
-1,609
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
30
Countdown
What is the expected sale value of a car with a Bluebook
value of $15,000?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
23
$12,590
$12,591
$12,592
$12,593
$12,594
$12,595
$12,596
$12,597
$12,598
$12,599
NonNon-Response Grid
30
Countdown
What is the expected sale value of a car with a Bluebook
value of $15,000?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
$12,590
$12,591
$12,592
$12,593
$12,594
$12,595
$12,596
$12,597
$12,598
$12,599
24
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
30
Countdown
What is the expected residual of a car with a Bluebook
value of $15,000?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
25
0
Root MSE
Coeff. of Det.
Std. Err. of Est.
Std. Err. of Slope
(Insufficient info)
NonNon-Response Grid
30
Countdown
What is the expected residual of a car with a Bluebook
value of $15,000?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
0
Root MSE
Coeff. of Det.
Std. Err. of Est.
Std. Err. of Slope
(Insufficient info)
26
0
0
0
0
0
0
NonNon-Response Grid
30
Countdown
What is the Root Mean Square Error?
(Note: Use N-2 in the denominator)
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
27
4,419.0
4,419.1
4,419.2
4,419.3
4,419.4
4,419.5
4,419.6
4,419.7
4,419.8
4,419.9
NonNon-Response Grid
30
Countdown
What is the Root Mean Square Error?
(Note: Use N-2 in the denominator)
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
4,419.0
4,419.1
4,419.2
4,419.3
4,419.4
4,419.5
4,419.6
4,419.7
4,419.8
4,419.9
28
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
30
Countdown
Assuming that these 5 observations represent the
population, what is the Coefficient of Determination?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
29
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
NonNon-Response Grid
60
Countdown
Assuming that these 5 observations represent the
population, what is the Coefficient of Determination?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
.70
.71
.72
.73
.74
.75
.76
.77
.78
.79
30
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
60
Countdown
Assuming that these 5 observations represent a sample,
what is the estimated Standard Error of the Slope?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
31
.300
.301
.302
.303
.304
.305
.306
.307
.308
.309
NonNon-Response Grid
30
Countdown
Assuming that these 5 observations represent a sample,
what is the estimated Standard Error of the Slope?
Auto
Number
1
2
3
4
5
Bluebk
Value
$7,429
$16,917
$20,946
$22,956
$26,219
Sold For
$3,708
$19,405
$18,562
$14,915
$25,078
0
1
2
3
4
5
6
7
8
9
.300
.301
.302
.303
.304
.305
.306
.307
.308
.309
32
0
0
0
0
0
0
0
0
0
0
NonNon-Response Grid
30
Countdown
The plot below represents part of a new dataset. Suppose that the
circled observation (located below the regression line) is located at the
mean of X. Removing it would ___ the intercept
33
0 Not change
1 Decrease
2 Increase
NonNon-Response Grid
30
Countdown
The plot below represents part of a new dataset. Suppose that the
circled observation (located below the regression line) is located at the
mean of X. Removing it would ___ the intercept
0 Not change
1 Decrease
2 Increase
34
0
0
0
NonNon-Response Grid
30
Countdown
35
Quiz Ends
36
From Populations to Samples
37
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
38
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
ocus is generally divided into two types of questions:
39
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
40
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
41
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
42
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
Accuracy in terms of specification error (as before)
43
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
Accuracy in terms of specification error (as before)
Accuracy in terms of sampling error
44
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
Accuracy in terms of specification error (as before)
Accuracy in terms of sampling error
Separating out the two types of error is challenging: how do you know what
you have left out?
45
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
Accuracy in terms of specification error (as before)
Accuracy in terms of sampling error
Separating out the two types of error is challenging: how do you know what
you have left out?
We can get a sense of how much error (specification and sampling) we are
making from the information contained in a single sample.
46
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
Accuracy in terms of specification error (as before)
Accuracy in terms of sampling error
Separating out the two types of error is challenging: how do you know what
you have left out?
We can get a sense of how much error (specification and sampling) we are
making from the information contained in a single sample.
This then leads us back to the two foci estimation and hypothesis testing in
the context of our estimate of the error
the context of our estimate of the error.
47
From Populations to Samples
Sample data are much more common in financial and econometric analysis
than are population data.
F
Estimation
Hypothesis Testing
Both types of questions revolve around the issue of accuracy
Accuracy in terms of specification error (as before)
Accuracy in terms of sampling error
Separating out the two types of error is challenging: how do you know what
you have left out?
We can get a sense of how much error (specification and sampling) we are
making from the information contained in a single sample.
This then leads us back to the two foci estimation and hypothesis testing in
the context of our estimate of the error
the context of our estimate of the error.
Describing the distribution of the slope under uncertainty: sampling
distribution of the slope
Estimating the Population Slope
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If we draw a sample from the population in order to
estimate the population slope, our best estimate of the
population slope is
1.
2.
3.
4.
5.
Sale Price
Price (Y)
48
the sample slope
the sample slope divided by the sample standard error
the standard error
the t-value times the standard error
a 95% confidence interval
$4,500,000
$4,000,000
$3,500,000
$3,000,000
$2,500,000
$2,000,000
$1,500,000
$1,000,000
$500,000
$0
0
2,000
4,000
6,000
Square Footage
8,000
NonNon-Response Grid
Estimating the Population Slope
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If we draw a sample from the population in order to
estimate the population slope, our best estimate of the
population slope is
1.
2.
3.
4.
5.
Sale Price
Price (Y)
49
the sample slope
the sample slope divided by the sample standard error
the standard error
the t-value times the standard error
a 95% confidence interval
0
0
0
0
0
$4,500,000
$4,000,000
$3,500,000
$3,000,000
$2,500,000
$2,000,000
$1,500,000
$1,000,000
$500,000
$0
0
2,000
4,000
6,000
Square Footage
8,000
NonNon-Response Grid
Sampling Error
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
50
If we draw a sample from the population in order to
estimate the population slope, the probability that the slope
of our sample is a long way away from the true population
slope is ____ the probability that the slope of our sample is
very close to the true population slope.
close to the true population slope
1.
2.
3.
4.
greater than
less than
equal to
to
(insufficient information to say)
Sampling Error
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
51
If we draw a sample from the population in order to
estimate the population slope, the probability that the slope
of our sample is a long way away from the true population
slope is ____ the probability that the slope of our sample is
very close to the true population slope.
close to the true population slope
1.
2.
3.
4.
greater than
less than
equal to
to
(insufficient information to say)
Sampling Distribution of the Slope
1
0
0
0
0
Sampling Error
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
52
If we draw a sample from the population in order to
estimate the population slope, the probability that the slope
of our sample is a long way away from the true population
slope is ____ the probability that the slope of our sample is
very close to the true population slope.
close to the true population slope
1.
2.
3.
4.
greater than
less than
equal to
to
(insufficient information to say)
0
0
0
0
Sampling Distribution of the Slope
If we drew lots of samples
If we drew lots of samples
from the population, the
slopes of those samples
would follow a t distribution
with n-2 degrees of freedom.
1
53
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
z
6.00
4.00
2.00
0.00
StandardErrors
2.00
4.00
6.00
54
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
z
6.00
4.00
2.00
0.00
StandardErrors
2.00
4.00
6.00
55
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
C
onverges to the normal distribution as the
onverges to the normal distribution as the
number of degrees of freedom approaches
infinity
z
6.00
4.00
2.00
0.00
StandardErrors
2.00
4.00
6.00
56
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
C
onverges to the normal distribution as the
onverges to the normal distribution as the
number of degrees of freedom approaches
infinity
Is typically employed instead of the normal
distribution when you have data from a
sample rather than from the entire
population
z
6.00
4.00
2.00
0.00
StandardErrors
2.00
4.00
6.00
57
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
C
onverges to the normal distribution as the
onverges to the normal distribution as the
number of degrees of freedom approaches
infinity
Is typically employed instead of the normal
distribution when you have data from a
sample rather than from the entire
population
z
6.00
4.00
2.00
0.00
2.00
4.00
StandardErrors
For example, when d.f.=6, a random
variable that is distributed as t will:
6.00
58
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
C
onverges to the normal distribution as the
onverges to the normal distribution as the
number of degrees of freedom approaches
infinity
Is typically employed instead of the normal
distribution when you have data from a
sample rather than from the entire
population
z
6.00
4.00
2.00
0.00
2.00
4.00
StandardErrors
For example, when d.f.=6, a random
variable that is distributed as t will:
t
ake on a value more positive than 2.45
standard errors 2.5% of the time
6.00
59
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
C
onverges to the normal distribution as the
onverges to the normal distribution as the
number of degrees of freedom approaches
infinity
Is typically employed instead of the normal
distribution when you have data from a
sample rather than from the entire
population
z
6.00
4.00
2.00
0.00
2.00
4.00
6.00
StandardErrors
For example, when d.f.=6, a random
variable that is distributed as t will:
t
take on a value more negative than 2.45
standard errors 2.5% of the time
60
The t-distribution
Looks very similar to the normal distribution
Slightly more squat
Slightly heavier tails
tdf=6
tdf=100
Is actually a family of distributions
The members of the family are identified
by the number of degrees of freedom
(d.f.)
C
onverges to the normal distribution as the
onverges to the normal distribution as the
number of degrees of freedom approaches
infinity
Is typically employed instead of the normal
di
distribution when you have data from a
sample rather than from the entire
population
z
6.00
4.00
2.00
0.00
2.00
4.00
6.00
StandardErrors
For example, when d.f.=6, a random
variable that is distributed as t will:
t
take on a value more negative than 2.45
standard errors 2.5% of the time
take on a value more positive than 2.45
standard errors or more negative than 2.45 standard errors 5% of the time.
[=tinv(.05,6)]
Standard Deviation of the Sampling Distribution
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If we draw a large number of samples from the population
and calculate the slope of each of those samples, the
standard deviation of the set of sample slopes is called the
1.
2.
3.
4.
population mean
population standard deviation
limit of central tendency
standard error
Sampling Distribution of the Slope
1
61
Standard Deviation of the Sampling Distribution
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
62
If we draw a large number of samples from the population
and calculate the slope of each of those samples, the
standard deviation of the set of sample slopes is called the
1.
2.
3.
4.
population mean
population standard deviation
limit of central tendency
standard error
0
0
0
0
Sampling Distribution of the Slope
An estimate of the
standard error of the
slope is provided as part
of the standard output
of the standard output
from Excel and SAS.
1
Upper Bound of a 95% Confidence Interval for the Slope
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If sample slopes were distributed normally, 25 samples out
of every 1000 will have a slope that is more than 1.96 (the
critical value) standard errors above the population
slope. But sample slopes are distributed as t, rather than
normally; what should the critical value be?
what should the critical value be?
1.
2.
3.
4.
1.96
a value which is less than 1.96
a value which is larger than 1.96
(not enough information provided to say)
Sampling Distribution of the Slope
1
63
Upper Bound of a 95% Confidence Interval for the Slope
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
64
If sample slopes were distributed normally, 25 samples out
of every 1000 will have a slope that is more than 1.96 (the
critical value) standard errors above the population
slope. But sample slopes are distributed as t, rather than
normally; what should the critical value be?
what should the critical value be?
1.
2.
3.
4.
1.96
a value which is less than 1.96
a value which is larger than 1.96
(not enough information provided to say)
Sampling Distribution of the Slope
The exact value can be
obtained by the
expression:
ti
tinv(.05, df)
df
in Excel, where df=n-2.
1
0
0
0
0
Lower Bound of a 95% Confidence Interval for the Slope
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
65
If sample slopes were distributed normally, 25 samples out
of every 1000 will have a slope that is lower than -1.96 (the
critical value) standard errors from the population slope.
But sample slopes are distributed as t, rather than
normally; what should the critical value be?
what should the critical value be?
1.
2.
3.
4.
-1.96
a value which is more negative than -1.96
a value which is less negative than -1.96
(not enough information provided to say)
Sampling Distribution of the Slope
1
Lower Bound of a 95% Confidence Interval for the Slope
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
66
If sample slopes were distributed normally, 25 samples out
of every 1000 will have a slope that is lower than -1.96 (the
critical value) standard errors from the population slope.
But sample slopes are distributed as t, rather than
normally; what should the critical value be?
what should the critical value be?
1.
2.
3.
4.
-1.96
a value which is more negative than -1.96
a value which is less negative than -1.96
(not enough information provided to say)
Sampling Distribution of the Slope
The exact value can be
obtained by the
expression:
-tinv(.05, df)
ti
df
in Excel, where df=n-2.
1
0
0
0
0
Upper Bound of a 95% Confidence Interval for the Slope:
Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If you repeatedly drew samples of size n=10, then 25 out of
every 1000 samples would have slopes that were more than
___ standard errors above the population slope.
1.
2.
3.
4.
4.
1.812
1.860
2.228
2.306
Sampling Distribution of the Slope
1
67
Upper Bound of a 95% Confidence Interval for the Slope:
Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If you repeatedly drew samples of size n=10, then 25 out of
every 1000 samples would have slopes that were more than
___ standard errors above the population slope.
1.
2.
3.
4.
4.
1.812
1.860
2.228
2.306
0
0
0
0
Sampling Distribution of the Slope
The exact value can be
obtained by the
expression:
ti
tinv(.05, 8)
8)
in Excel.
1
68
Upper Bound of a 95% Confidence Interval for the Slope:
Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
SqFt (X)
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If you repeatedly drew samples of size n=10, then 25 out of
every 1000 samples would have slopes that were more than
___ standard errors above the population slope.
1.
2.
3.
4.
4.
1.812
1.860
2.228
2.306
0
0
0
0
Sampling Distribution of the Slope
The exact value can be
obtained by the
expression:
ti
tinv(.05, 8)
8)
in Excel.
1
The equivalent of tinv(.05,8) can be obtained by consulting Gujarati
Table D2, row labelled df=8, column labelled Pr=.025/.05
69
Lower Bound of a 95% Confidence Interval for the Slope:
Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If you repeatedly drew samples of size n=10, then 25 out of
every 1000 samples would have slopes that are lower than
___ standard errors below the population slope.
1.
2.
3.
4.
4.
-1.812
-1.860
-2.228
-2.306
Sampling Distribution of the Slope
1
70
Lower Bound of a 95% Confidence Interval for the Slope:
Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If you repeatedly drew samples of size n=10, then 25 out of
every 1000 samples would have slopes that are lower than
___ standard errors below the population slope.
1.
2.
3.
4.
4.
-1.812
-1.860
-2.228
-2.306
0
0
0
0
Sampling Distribution of the Slope
The exact value can be
obtained by the
expression:
-tinv(.05, 8)
ti
8)
in Excel.
1
71
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
72
If you repeatedly drew samples of size n=10, then ___ out of
every 1000 samples would have slopes that are more than
2.306 standard errors away (in either direction) from the
population slope.
1.
2.
3.
4.
Less than 25
Less than 50
At least 50
More than 50
than 50
Sampling Distribution of the Slope
1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
73
If you repeatedly drew samples of size n=10, then ___ out of
every 1000 samples would have slopes that are more than
2.306 standard errors away (in either direction) from the
population slope.
1.
2.
3.
4.
Less than 25
Less than 50
At least 50
More than 50
than 50
0
0
0
0
Sampling Distribution of the Slope
The exact number,
expressed as a proportion,
can be obtained by the
Excel function:
=tdist(2.306,df,2)
where df=n-2.
1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If you drew a single sample of size n=10, then what is the
th
th
probability that the true population slope is more than
2.306 standard errors away from the slope of that sample?
1.
2.
3.
4.
Less than .025
Less than .05
At least .05
More than .05
Sampling Distribution of the Slope
b1
74
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 2
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
75
If you drew a single sample of size n=10, then what is the
th
th
probability that the true population slope is more than
2.306 standard errors away from the slope of that sample?
1.
2.
3.
4.
Less than .025
Less than .05
At least .05
More than .05
0
0
0
0
Sampling Distribution of the Slope
The exact proportion can
be obtained by the Excel
function:
=tdist(2.306,df,2)
where df=n-2.
b1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 3
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then what is the probability that the
population slope is between 534 and 766?
1.
2.
3.
4.
More than .97
More than .96
More than .95
.95 or less
Sampling Distribution of the Slope
b1
76
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 3
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then what is the probability that the
population slope is between 534 and 766?
1.
2.
3.
4.
More than .97
More than .96
More than .95
.95 or less
0
0
0
0
Sampling Distribution of the Slope
1. (766-650)/50= 2.32
(534-650)/50=-2.32
2. tdist(2.32,8,2)=.0489
3. 1-.0489=.9511
b1
77
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 3
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then what is the probability that the
population slope is between 534 and 766?
1.
2.
3.
4.
More than .97
More than .96
More than .95
.95 or less
0
0
0
0
Sampling Distribution of the Slope
1. (766-650)/50= 2.32
(534-650)/50=-2.32
2. tdist(2.32,8,2)=.0489
3. 1-.0489=.9511
b1
78
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 3
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then what is the probability that the
population slope is between 534 and 766?
1.
2.
3.
4.
More than .97
More than .96
More than .95
.95 or less
0
0
0
0
Sampling Distribution of the Slope
1. (766-650)/50= 2.32
(534-650)/50=-2.32
2. tdist(2.32,8,2)=.0489
3. 1-.0489=.9511
b1
79
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 3
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then what is the probability that the
population slope is between 534 and 766?
1.
2.
3.
4.
More than .97
More than .96
More than .95
.95 or less
0
0
0
0
Sampling Distribution of the Slope
1. (766-650)/50= 2.32
(534-650)/50=-2.32
2. tdist(2.32,8,2)=.0489
3. 1-.0489=.9511
b1
80
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 4
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{592, 708}
{564, 736}
{534, 765}
{506, 794}
Sampling Distribution of the Slope
b1
81
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 4
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
82
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{592, 708}
{564, 736}
{534, 765}
{506, 794}
0
0
0
0
Sampling Distribution of the Slope
1. tinv(.05,8)=2.306004133
2. 650+2.306004133*50 =
765.30
650-2.306004133*50=
534.70
b1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 4
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
83
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{592, 708}
{564, 736}
{534, 765}
{506, 794}
0
0
0
0
Sampling Distribution of the Slope
1. tinv(.05,8)=2.306004133
2. 650+2.306004133*50 =
765.30
650-2.306004133*50=
534.70
b1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Round 4
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
84
If , based on a single sample of size n=10, you estimate the
th
population slope to be 650 and the standard error of the
slope to be 50, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{592, 708}
{564, 736}
{534, 765}
{506, 794}
0
0
0
0
Sampling Distribution of the Slope
1. tinv(.05,8)=2.306004133
2. 650+2.306004133*50 =
765.30
650-2.306004133*50=
534.70
b1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Retry
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
If , based on a single sample of size n=12, you estimate the
th
population slope to be 500 and the standard error of the
slope to be 40, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{410, 589}
{411, 589}
{411, 590}
{410, 590}
Sampling Distribution of the Slope
b1
85
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Retry
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
86
If , based on a single sample of size n=12, you estimate the
th
population slope to be 500 and the standard error of the
slope to be 40, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{410, 589}
{411, 589}
{411, 590}
{410, 590}
0
0
0
0
Sampling Distribution of the Slope
1. tinv(.05,10)=2.22813884
2. 500+2.22813884*40 =
589.13
500-2.22813884*40 =
410
410.87
b1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Retry
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
87
If , based on a single sample of size n=12, you estimate the
th
population slope to be 500 and the standard error of the
slope to be 40, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{410, 589}
{411, 589}
{411, 590}
{410, 590}
0
0
0
0
Sampling Distribution of the Slope
1. tinv(.05,10)=2.22813884
2. 500+2.22813884*40 =
589.13
500-2.22813884*40 =
410
410.87
b1
Lower and Upper Bounds of a 95% Confidence Interval
for the Slope: Retry
Price (Y)
1
2
3
4
5
6
7
8
9
10
11
11
12
13
14
15
16
17
18
19
20
20
$635,000
$815,000
$885,000
$784,000
$879,000
$1,060,000
$980,000
$785,000
$1,385,000
$950,000
$980,000
$1,125,000
$1,150,000
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,580
1,617
1,740
1,896
1,940
1,947
2,062
2,186
2,223
2,380
2,501
2,634
3,125
3,150
3,280
3,524
3,977
5,526
6,370
88
If , based on a single sample of size n=12, you estimate the
th
population slope to be 500 and the standard error of the
slope to be 40, then between what two values can we be
more than 95% confident that the population mean falls?
1.
2.
3.
4.
{410, 589}
{411, 589}
{411, 590}
{410, 590}
0
0
0
0
Sampling Distribution of the Slope
1. tinv(.05,10)=2.22813884
2. 500+2.22813884*40 =
589.13
500-2.22813884*40 =
410
410.87
b1
Hypothesis Testing: Introduction and Groundrules
89
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2.
90
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
91
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
3. The H0 and HA always refer to the population, not to the sample.
4.
92
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
3. The H0 and HA always refer to the population, not to the sample.
4. The logic of hypothesis testing allows us only to disconfirm H0 (at the
stipulated level of confidence), not to confirm it.
Analogously, the logic of hypothesis testing allows us only to confirm HA (at
the stipulated level of confidence), not to disconfirm it.
the stipulated level of confidence), not to disconfirm it.
93
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
3. The H0 and HA always refer to the population, not to the sample.
4. The logic of hypothesis testing allows us only to disconfirm H0 (at the
stipulated level of confidence), not to confirm it.
Analogously, the logic of hypothesis testing allows us only to confirm HA (at
the stipulated level of confidence), not to disconfirm it.
the stipulated level of confidence), not to disconfirm it.
5. H0 must always contain the equal sign.
6.
94
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
3. The H0 and HA always refer to the population, not to the sample.
4. The logic of hypothesis testing allows us only to disconfirm H0 (at the
stipulated level of confidence), not to confirm it.
Analogously, the logic of hypothesis testing allows us only to confirm HA (at
the stipulated level of confidence), not to disconfirm it.
the stipulated level of confidence), not to disconfirm it.
5. H0 must always contain the equal sign.
e.g., 1=0 is ok as a H0; 10 is not .
95
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
3. The H0 and HA always refer to the population, not to the sample.
4. The logic of hypothesis testing allows us only to disconfirm H0 (at the
stipulated level of confidence), not to confirm it.
Analogously, the logic of hypothesis testing allows us only to confirm HA (at
the stipulated level of confidence), not to disconfirm it.
the stipulated level of confidence), not to disconfirm it.
5. H0 must always contain the equal sign.
e.g., 1=0 is ok as a H0; 10 is not .
1. A logical consequence is that certain hypotheses are unconfirmable: e.g.,
there is no linear relationship between square footage and sale price is
NOT confirmable.
2.
96
Hypothesis Testing: Introduction and Groundrules
1. Hypothesis testing allows you to test an assertion about the population. For
example, In the population, Sales Price is linearly related to Square
Footage.
2. Hypothesis testing begins with the specification of a null hypothesis (H0)
which you wish to disconfirm, and its obverse (the alternative hypothesis,
hi
di
it
(th
HA) which you wish to confirm.
3. The H0 and HA always refer to the population, not to the sample.
4. The logic of hypothesis testing allows us only to disconfirm H0 (at the
stipulated level of confidence), not to confirm it.
Analogously, the logic of hypothesis testing allows us only to confirm HA (at
the stipulated level of confidence), not to it.
the disconfirm stipulated level of confidence), not to disconfirm it.
5. H0 must always contain the equal sign.
e.g., 1=0 is ok as a H0; 10 is not .
1. A logical consequence is that certain hypotheses are unconfirmable: e.g.,
there is no linear relationship between square footage and sale price is
NOT confirmable.
2. Either of two analytical approaches can be used to conduct a hypothesis test:
of two analytical appro
can be used to conduct hypothesis test:
a confidence interval approach, and a test-of-significance approach.
97
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
98
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
99
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
100
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
stablish a requisite confidence level
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
101
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
By convention, usually (but not always) 95%
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
102
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
By convention, usually (but not always) 95%
2. Estimate the population slope, standard error of the slope from a random
sample
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
103
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
By convention, usually (but not always) 95%
2. Estimate the population slope, standard error of the slope from a random
sample
1. Calculate the lower and upper bounds on a confidence interval around the
slope
slope
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
104
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
By convention, usually (but not always) 95%
2. Estimate the population slope, standard error of the slope from a random
sample
1. Calculate the lower and upper bounds on a confidence interval around the
slope
slope
3. Determine whether zero is contained within the interval
4.
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
105
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
By convention, usually (but not always) 95%
2. Estimate the population slope, standard error of the slope from a random
sample
1. Calculate the lower and upper bounds on a confidence interval around the
slope
slope
3. Determine whether zero is contained within the interval
If not, reject H0 and conclude that H0 is false;
equivalently, accept HA and conclude that HA is true
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
106
A Confidence Interval Approach to Hypothesis Testing
1. Establish the null hypothesis (H0) and alternative hypothesis (HA)1
H0: In the population, square footage and sale price are linearly unrelated
(1=0)
HA: In the population, square footage and sale price are linearly related
In the population square footage and sale price are linearly related
(10)
1. E
By convention, usually (but not always) 95%
2. Estimate the population slope, standard error of the slope from a random
sample
1. Calculate the lower and upper bounds on a confidence interval around the
slope
slope
3. Determine whether zero is contained within the interval
If not, reject H0 and conclude that H0 is false;
equivalently, accept HA and conclude that HA is true
If so, fail to reject H0 and fail to conclude that H0 is false; equivalently,
fail to accept HA and fail to conclude that HA is true
1Some
textbook authors, including Gujarati, use H1 to designate the alternative hypothesis,
rather than HA. In the course, we follow the more common practice of using HA to designate
the alternative hypothesis.
107
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
108
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
109
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
110
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
111
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
UB: 550+2.0484*285=1133.796
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
112
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
UB: 550+2.0484*285=1133.796
95% confident that 1 is
contained somewhere in the interval
contained somewhere in the interval
{-33.796, 1133.796}
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
113
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
UB: 550+2.0484*285=1133.796
95% confident that 1 is
contained somewhere in the interval
contained somewhere in the interval
{-33.796, 1133.796}
Because the confidence interval
contains 0:
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
114
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
UB: 550+2.0484*285=1133.796
95% confident that 1 is
contained somewhere in the interval
contained somewhere in the interval
{-33.796, 1133.796}
Because the confidence interval
contains 0:
Fail to reject H0
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
115
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
UB: 550+2.0484*285=1133.796
95% confident that 1 is
contained somewhere in the interval
contained somewhere in the interval
{-33.796, 1133.796}
Because the confidence interval
contains 0:
Fail to reject H0
Fail to accept HA
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
116
0
0
0
Hypothesis Testing: Confidence Interval Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 550 and a standard error of 285. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
LB: 550-2.0484*285=-33.796
UB: 550+2.0484*285=1133.796
95% confident that 1 is
contained somewhere in the interval
contained somewhere in the interval
{-33.796, 1133.796}
Because the confidence interval
contains 0:
Fail to reject H0
Fail to accept HA
Fail to conclude with 95% confidence
that Price is predictable from Square Feet in the
that Price is predictable from Square Feet in the
population.
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
117
0
0
0
Hypothesis Testing (Confidence Interval Approach): Retry
These eight houses represent a random sample from the
population of neighboring recently
population of neighboring recently-sold houses. Is there
houses Is there
reasonable (say, 95%) certainty that price is linearly related
to square feet in the population?
Sqft
1,191
1,580
1,701
2,083
2,527
2,684
2,912
3,670
1. No
2. Yes
3. Not enough
information
Price
842,310
867,727
878,798
927,203
912,795
817,547
909,932
957,883
^ =32.59471
1
s =19.17610
1
118
Hypothesis Testing (Confidence Interval Approach): Retry
These eight houses represent a random sample from the
population of neighboring recently
population of neighboring recently-sold houses. Is there
houses Is there
reasonable (say, 95%) certainty that price is linearly related
to square feet in the population?
Sqft
1,191
1,580
1,701
2,083
2,527
2,684
2,912
3,670
1. No
2. Yes
3. Not enough
information
Price
842,310
867,727
878,798
927,203
912,795
817,547
909,932
957,883
^ =32.59471
1
s =19.17610
1
119
0
0
0
Why/why not?
Hypothesis Testing (Confidence Interval Approach): Retry
These eight houses represent a random sample from the
population of neighboring recently
population of neighboring recently-sold houses. Is there
houses Is there
reasonable (say, 95%) certainty that price is linearly related
to square feet in the population?
Sqft
1,191
1,580
1,701
2,083
2,527
2,684
2,912
3,670
1. No
2. Yes
3. Not enough
information
Price
842,310
867,727
878,798
927,203
912,795
817,547
909,932
957,883
^ =32.59471
1
s =19.17610
1
120
0
0
0
Why/why not?
tinv(.05,6) = 2.44691
c.i.slope: {-14.33,79.52}
Hypothesis Testing (Confidence Interval Approach): Retry
These eight houses represent a random sample from the
population of neighboring recently
population of neighboring recently-sold houses. Is there
houses Is there
reasonable (say, 95%) certainty that price is linearly related
to square feet in the population?
$1,000,000
Sqft
1,191
1,580
1,701
2,083
2,527
2,684
2,912
3,670
$950,000
Price
$900,000
$850,000
$800,000
$750,000
0
1,000 2,000 3,000 4,000 5,000 6,000 7,000
SquareFeet
1. No
2. Yes
3. Not enough
information
Price
842,310
867,727
878,798
927,203
912,795
817,547
909,932
957,883
^ =32.59471
1
s =19.17610
1
121
0
0
0
Why/why not?
tinv(.05,6) = 2.44691
c.i.slope: {-14.33,79.52}
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
122
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
123
0
0
0
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
124
0
0
0
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
t=(600-0)/250=2.40
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
125
0
0
0
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
t=(600-0)/250=2.40
Because 2.40>2.0484:
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
126
0
0
0
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
t=(600-0)/250=2.40
Because 2.40>2.0484:
Reject H0
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
127
0
0
0
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
t=(600-0)/250=2.40
Because 2.40>2.0484:
Reject H0
Accept
Accept HA
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
128
0
0
0
Hypothesis Testing: Test-of-Significance Approach
We wish to know whether, in the population, Price is at all
th
predictable from Square Feet. A n=30 random sample reveals
a sample slope of 600 and a standard error of 250. Can we be
95% confident that Price is predictable from Square Feet in
the population?
H0: 1=0
HA: 10
tcrit=tinv(.05,30-2)=2.0484
t=(600-0)/250=2.40
Because 2.40>2.0484:
Reject H0
Accept
Accept HA
Conclude with 95% confidence
that Price is predictable from
Square Feet in the population.
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
129
0
0
0
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
130
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
131
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
132
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
0: 1=0 The slope of the best-fitting straight line is zero in the pop.
Th
th
li
th
133
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
HA: 10 The slope of the best-fitting straight line is not zero in the pop.
Th
th
li
th
134
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
HA: 10 The slope of the best-fitting straight line is not zero in the pop.
Th
th
li
th
1. If you wish to test the hypothesis that Higher Square Footage is associated
with higher Prices in the population:
135
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
HA: 10 The slope of the best-fitting straight line is not zero in the pop.
Th
th
li
th
1. If you wish to test the hypothesis that Higher Square Footage is associated
with higher Prices in the population:
H0: 10 The slope of the best-fitting straight line is non-positive in the pop.
136
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
HA: 10 The slope of the best-fitting straight line is not zero in the pop.
Th
th
li
th
1. If you wish to test the hypothesis that Higher Square Footage is associated
with higher Prices in the population:
H0: 10 The slope of the best-fitting straight line is non-positive in the pop.
HA: 1>0 The slope of the best-fitting straight line is positive in the pop.
137
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
HA: 10 The slope of the best-fitting straight line is not zero in the pop.
Th
th
li
th
1. If you wish to test the hypothesis that Higher Square Footage is associated
with higher Prices in the population:
H0: 10 The slope of the best-fitting straight line is non-positive in the pop.
HA: 1>0 The slope of the best-fitting straight line is positive in the pop.
1. It is important to specify a hypothesis as directional when it is appropriate to do
so: doing so provides you with more statistical power.
138
Directional Alternative Hypotheses
1. There are times when, rather than wishing to test a hypothesis that a
relationship exists between two variables (e.g., Price is predictable from
Square Feet in the population), we wish to test that a relationship exists in a
particular direction (e.g., In the population, the more Square Feet a house has,
the greater its Price)
the greater its Price).
2. There is a qualitative difference between these two types of hypotheses, and
they are neither specified nor tested in exactly the same way.
3. If you wish to test the hypothesis that Price is predictable from Square Feet in
th
th P
the population:
H
HA: 10 The slope of the best-fitting straight line is not zero in the pop.
Th
th
li
th
1. If you wish to test the hypothesis that Higher Square Footage is associated
with higher Prices in the population:
H0: 10 The slope of the best-fitting straight line is non-positive in the pop.
HA: 1>0 The slope of the best-fitting straight line is positive in the pop.
1. It is important to specify a hypothesis as directional when it is appropriate to do
so: doing so provides you with more statistical power.
1. In order to test a directional alternative hypothesis, you must use the testof-significance approach, not the confidence interval approach
139
Guidelines for Specifying Directional Hypotheses
140
Guidelines for Specifying Directional Hypotheses
Th
The focus should always be on the alternative hypothesis, because that
th
th
is what you are attempting to conclude is true.
141
Guidelines for Specifying Directional Hypotheses
Th
The focus should always be on the alternative hypothesis, because that
th
th
is what you are attempting to conclude is true.
T
he equality symbol must always appear in the null hypothesis (e.g.,
or ) and may never appear in the alternative hypothesis
(which, in the case of directional hypotheses, must always be > or
<)
<).
142
Guidelines for Specifying Directional Hypotheses
Th
The focus should always be on the alternative hypothesis, because that
th
th
is what you are attempting to conclude is true.
T
Textbook authors tend to vary in whether or not the directional symbol
is included in the null hypothesis. Thus, the reader has to be clear that
the following two specifications are fully equivalent:
<).
H0: 2 0
Ha: 2 > 0
H0: 2 = 0
Ha: 2 > 0
143
The Rationale for Specifying Directional Hypotheses
144
The Rationale for Specifying Directional Hypotheses
The power of a statistical test to detect a correct alternative
power of statistical test to detect correct alternative
hypothesis is increased when a directional alternative hypothesis is
specified and tested.
145
The Rationale for Specifying Directional Hypotheses
The power of a statistical test to detect a correct alternative
power of statistical test to detect correct alternative
hypothesis is increased when a directional alternative hypothesis is
specified and tested.
Power is defined as the ability of the test to lead to a conclusion to
Power is defined as the ability of the test to lead to conclusion to
reject the null hypothesis when, in fact, the null hypothesis is false.
146
The Rationale for Specifying Directional Hypotheses
The power of a statistical test to detect a correct alternative
power of statistical test to detect correct alternative
hypothesis is increased when a directional alternative hypothesis is
specified and tested.
Power is defined as the ability of the test to lead to a conclusion to
Power is defined as the ability of the test to lead to conclusion to
reject the null hypothesis when, in fact, the null hypothesis is false.
F
ramed (equivalently) in terms of the alternative hypothesis, power
is defined as the ability of the test to lead to a conclusion to accept the
alternative hypothesis when, in fact, the alternative hypothesis is true.
147
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
148
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
149
0
0
0
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
tcrit=tinv(.10,30-2)=1.7011
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
150
0
0
0
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
tcrit=tinv(.10,30-2)=1.7011
t=(550-0)/300=1.83
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
151
0
0
0
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
tcrit=tinv(.10,30-2)=1.7011
t=(550-0)/300=1.83
Because 1.83>1.70:
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
152
0
0
0
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
tcrit=tinv(.10,30-2)=1.7011
t=(550-0)/300=1.83
Because 1.83>1.70:
Reject H0
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
153
0
0
0
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
tcrit=tinv(.10,30-2)=1.7011
t=(550-0)/300=1.83
Because 1.83>1.70:
Reject H0
Accept
Accept HA
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
154
0
0
0
Hypothesis Testing: Directional Alternative Hypothesis
We wish to know whether, in the population, Price is
th
positively related to Square Feet. A n=30 random sample
reveals a sample slope of 550 and a standard error of 300.
Can we be 95% confident that Price is positively related to
Square Feet in the population?
H0: 10
HA: 1>0
tcrit=tinv(.10,30-2)=1.7011
t=(550-0)/300=1.83
Because 1.83>1.70:
Reject H0
Accept
Accept HA
Conclude with 95% confidence
that Price is positively related to
Square Feet in the population.
1. No
2. Yes
3. Not enough
information
Sampling Distribution of the Slope
b1
155
0
0
0
These eight houses represent a random sample from the population of
neighboring recently-sold houses. Is there reasonable (say, 95%)
certainty that price is linearly UNrelated to square feet in the population?
th
li
UN
th
$1,000,000
156
0 No
1 Yes
2 Unable to say
$950,000
Sqft
1,191
1,580
1,701
2,083
2,527
2,684
2,912
3,670
Price
$900,000
$850,000
$800,000
$750,000
0
1,000 2,000 3,000 4,000 5,000 6,000 7,000
SquareFeet
Price
842,310
867,727
878,798
927,203
912,795
817,547
909,932
957,883
^ =32.59471
1
s =19.17610
1
NonNon-Response Grid
tinv(.05,6) = 2.44691
c.i.slope: {-14.33,79.52}
These eight houses represent a random sample from the population of
neighboring recently-sold houses. Is there reasonable (say, 95%)
certainty that price is linearly UNrelated to square feet in the population?
th
li
UN
th
$1,000,000
0 No
1 Yes
2 Unable to say
$950,000
Sqft
1,191
1,580
1,701
2,083
2,527
2,684
2,912
3,670
Price
$900,000
$850,000
$800,000
$750,000
0
1,000 2,000 3,000 4,000 5,000 6,000 7,000
SquareFeet
1
0
0
0
Price
842,310
867,727
878,798
927,203
912,795
817,547
909,932
957,883
^ =32.59471
1
s =19.17610
157
Why/why not?
NonNon-Response Grid
tinv(.05,6) = 2.44691
c.i.slope: {-14.33,79.52}
Alternative to the Classical Procedure for
Reaching a Conclusion
158
Alternative to the Classical Procedure for
Reaching a Conclusion
A considerable disadvantage of the classical procedure for reaching a
conclusion lies in the necessity of obtaining the critical value (e.g.,
ti
tinv in Excel or a table in the back of Gujarati).
th
159
Alternative to the Classical Procedure for
Reaching a Conclusion
A considerable disadvantage of the classical procedure for reaching a
conclusion lies in the necessity of obtaining the critical value (e.g.,
ti
tinv in Excel or a table in the back of Gujarati).
th
Many statistical computing packages provide a p-value as an
alternative (and fully equivalent) mechanism for reaching a conclusion
about whether or not to reject the null hypothesis that
about whether or not to reject the null hypothesis that 1 = 0.
160
Alternative to the Classical Procedure for
Reaching a Conclusion
A considerable disadvantage of the classical procedure for reaching a
conclusion lies in the necessity of obtaining the critical value (e.g.,
ti
tinv in Excel or a table in the back of Gujarati).
th
Many statistical computing packages provide a p-value as an
alternative (and fully equivalent) mechanism for reaching a conclusion
about whether or not to reject the null hypothesis that
about whether or not to reject the null hypothesis that 1 = 0.
To use the p-value, you see whether the p-value associated with b1 is
less than the -level you set prior to collecting any data. If it is, you
reject H0 (or, equivalently, you accept Ha); otherwise you fail to
reject H0 (or, equivalently, you fail to accept Ha).
161
Alternative to the Classical Procedure for
Reaching a Conclusion
A considerable disadvantage of the classical procedure for reaching a
conclusion lies in the necessity of obtaining the critical value (e.g.,
ti
tinv in Excel or a table in the back of Gujarati).
th
Many statistical computing packages provide a p-value as an
alternative (and fully equivalent) mechanism for reaching a conclusion
about whether or not to reject the null hypothesis that
about whether or not to reject the null hypothesis that 1 = 0.
To use the p-value, you see whether the p-value associated with b1 is
less than the -level you set prior to collecting any data. If it is, you
reject H0 (or, equivalently, you accept Ha); otherwise you fail to
reject H0 (or, equivalently, you fail to accept Ha).
Formally, the p-value is defined as the probability that you would be
making a Type I error if you rejected the null hypothesis based on the
results from your sample.
lt
162
P-values
In Excel:
In SAS:
proc glm data=mf1;
model price = sqft;
run; quit;
163
The Two-Step Rule for Directional Alternative Hypotheses (1):
Reject H0? Why
Price (Y)
$635,000
$885
$885,000
$1,385,000
$1,225,000
$1,059,000
$3,250,000
$3,275,000
$3,995,000
H0: 10
HA: 1>0
164
SqFt (X)
1,290
1,617
2,186
3,125
3,150
3,977
5,526
6,370
Step 1: Divide p in half. Is the result less than ?
Step 2: Is the sample slope consistent in
directionality with Ha?
If (and only if) both of these conditions are met,
then (and only then) you can reject H0 in favor of
the directional HA.
0 No
1 Yes
NonNon-Response Grid
The Two-Step Rule for Directional Alternative Hypotheses (1):
Reject H0? Why
Price (Y)
$635,000
$885
$885,000
$1,385,000
$1,225,000
$1,059,000
$3,250,000
$3,275,000
$3,995,000
H0: 10
HA: 1>0
165
SqFt (X)
1,290
1,617
2,186
3,125
3,150
3,977
5,526
6,370
Step 1: Divide p in half. Is the result less than ?
Step 2: Is the sample slope consistent in
directionality with Ha?
If (and only if) both of these conditions are met,
then (and only then) you can reject H0 in favor of
the directional HA.
0 No
1 Yes
0
0
Why/?
NonNon-Response Grid
The Two-Step Rule for Directional Alternative Hypotheses (2):
Reject H0? Why
Price (Y)
$635,000
$885,000
SqFt (X)
1,290
1,617
$1,385,000
$1,225,000
$1,059,000
2,186
3,125
3,150
166
H0: 10
HA: 1>0
Step 1: Divide p in half. Is the result less than ?
Step 2: Is the sample slope consistent in
di
directionality with Ha?
If (and only if) both of these conditions are met,
then (and only then) you can reject H0 in favor of
the directional HA.
0 No
1 Yes
0
0
Why/?
NonNon-Response Grid
The Two-Step Rule for Directional Alternative Hypotheses (2):
Reject H0? Why
Price (Y)
$635,000
$885,000
SqFt (X)
1,290
1,617
$1,385,000
$1,225,000
$1,059,000
2,186
3,125
3,150
167
H0: 10
HA: 1>0
Step 1: Divide p in half. Is the result less than ?
Step 2: Is the sample slope consistent in
di
directionality with Ha?
If (and only if) both of these conditions are met,
then (and only then) you can reject H0 in favor of
the directional HA.
0 No
1 Yes
0
0
Why/?
NonNon-Response Grid
Prediction Interval Bands
(Based On 7-House Sample)
168
Prediction Interval Bands
(Based On 7-House Sample)
169
Prediction Interval Bands
(Based On 7-House Sample)
$3754K
170
Prediction Interval Bands
(Based On 7-House Sample)
$3754K
$690K
171
Prediction Interval Width
Intuitively, what do you think is going to happen to the
thi
th
prediction interval bands if you increase the sample size?
n=7
172
0 Nothing
1 Closer together
2 Farther apart
Prediction Interval Width
Intuitively, what do you think is going to happen to the
thi
th
prediction interval bands if you increase the sample size?
n=7
173
0 Nothing
1 Closer together
2 Farther apart
n=100
0
0
0
Adjusted R-Squared
174
A random sample of seven recently-sold high-end houses is selected for
hi
study. If r2 for the sample is 0.810541, then the value defined by the
algebraic equation below is
n 1
1 (1 r )
n2
2
0
1
2
3
4
5
6
7
8
9
0.70
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
Price (Y)
SqFt (X)
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
3,125
3,150
3,280
3,524
3,977
5,526
6,370
Adjusted R-Squared
175
A random sample of seven recently-sold high-end houses is selected for
hi
study. If r2 for the sample is 0.810541, then the value defined by the
algebraic equation below is
n 1
1 (1 r )
n2
2
0
1
2
3
4
5
6
7
8
9
0.70
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0
0
0
0
0
0
0
0
0
0
Price (Y)
Adjusted r2: Our best estimate of the population coefficient
of determination when the data come from a
sample.
Depending on which SAS Proc you use, you
ma ha to comp
may have to compute this by hand.
this
hand
SqFt (X)
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
3,125
3,150
3,280
3,524
3,977
5,526
6,370
Adjusted R-Squared
176
A random sample of seven recently-sold high-end houses is selected for
hi
study. If r2 for the sample is 0.810541, then the value defined by the
algebraic equation below is
n 1
1 (1 r )
n2
2
Note: This number represents the number of parameters we
are estimating. As we develop more sophisticated
models, this number will get larger.
Adjusted r2: Our best estimate of the population coefficient
of determination when the data come from a
sample.
Depending on which SAS Proc you use, you
ma ha to comp
may have to compute this by hand.
this
hand
0
1
2
3
4
5
6
7
8
9
0.70
0.71
0.72
0.73
0.74
0.75
0.76
0.77
0.78
0.79
0
0
0
0
0
0
0
0
0
0
Price (Y)
SqFt (X)
$1,225,000
$1,059,000
$1,450,000
$2,075,000
$3,250,000
$3,275,000
$3,995,000
3,125
3,150
3,280
3,524
3,977
5,526
6,370
Adjusted R-Squared: Retry
A different random sample of seven recently-sold high-end houses is
diff
hi
selected for study. If r2 for the sample is 0.855685, then the value
defined by the algebraic equation below is
n 1
1 (1 r )
n2
2
177
0
1
2
3
4
5
6
7
8
9
0.77
0.78
0.79
0.80
0.81
0.82
0.83
0.84
0.85
0.86
Price (Y)
$635,000
$885,000
$1,385,000
$1,225,000
$1,059,000
$3,250,000
$3,275,000
$3,995,000
SqFt (X)
1,290
1,617
2,186
3,125
3,150
3,977
5,526
6,370
Adjusted R-Squared: Retry
A different random sample of seven recently-sold high-end houses is
diff
hi
selected for study. If r2 for the sample is 0.855685, then the value
defined by the algebraic equation below is
n 1
1 (1 r )
n2
2
Adjusted r2: Our best estimate of the population coefficient
of determination when the data come from a
sample.
178
0
1
2
3
4
5
6
7
8
9
0.77
0.78
0.79
0.80
0.81
0.82
0.83
0.84
0.85
0.86
Price (Y)
$635,000
$885,000
$1,385,000
$1,225,000
$1,059,000
$3,250,000
$3,275,000
$3,995,000
0
0
0
0
0
0
0
0
0
0
SqFt (X)
1,290
1,617
2,186
3,125
3,150
3,977
5,526
6,370
Testing Hypotheses About 0
179
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
180
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
181
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
This value represents our best estimate of the standard deviation of b0 if we
were to draw many samples from the population.
182
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
This value represents our best estimate of the standard deviation of b0 if we
were to draw many samples from the population.
The calculation is sufficiently complex that it is best left to the computer.
calc
is
comple that it is best left to the comp
183
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
This value represents our best estimate of the standard deviation of b0 if we
were to draw many samples from the population.
The calculation is sufficiently complex that it is best left to the computer.
calc
is
comple that it is best left to the comp
Once we have b0 and sb0 from our sample, statistical theory tells us that
(given the assumptions we have discussed), b0/sb0 (t) will be distributed
as a t distribution with n-p degrees of freedom, where n is the number
of observations and p is the number of parameters in the model.
184
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
This value represents our best estimate of the standard deviation of b0 if we
were to draw many samples from the population.
The calculation is sufficiently complex that it is best left to the computer.
calc
is
comple that it is best left to the comp
Once we have b0 and sb0 from our sample, statistical theory tells us that
(given the assumptions we have discussed), b0/sb0 (t) will be distributed
as a t distribution with n-p degrees of freedom, where n is the number
of observations and p is the number of parameters in the model.
We can compare t with the critical value of t (at a given level of ) in
order to determine whether or not to reject the null hypothesis.
185
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
This value represents our best estimate of the standard deviation of b0 if we
were to draw many samples from the population.
The calculation is sufficiently complex that it is best left to the computer.
calc
is
comple that it is best left to the comp
Once we have b0 and sb0 from our sample, statistical theory tells us that
(given the assumptions we have discussed), b0/sb0 (t) will be distributed
as a t distribution with n-p degrees of freedom, where n is the number
of observations and p is the number of parameters in the model.
We can compare t with the critical value of t (at a given level of ) in
order to determine whether or not to reject the null hypothesis.
Conversely, we can use the printed p-value to make the determination.
186
Testing Hypotheses About 0
Although the discussion thus far has focused on testing hypotheses
oug
about 1, exactly the same process can be applied to hypotheses about
0:
Given the assumptions, it is possible to compute an estimate of the
standard error of the intercept, sb0, from a single sample.
This value represents our best estimate of the standard deviation of b0 if we
were to draw many samples from the population.
The calculation is sufficiently complex that it is best left to the computer.
calc
is
comple that it is best left to the comp
Once we have b0 and sb0 from our sample, statistical theory tells us that
(given the assumptions we have discussed), b0/sb0 (t) will be distributed
as a t distribution with n-p degrees of freedom, where n is the number
of observations and p is the number of parameters in the model.
We can compare t with the critical value of t (at a given level of ) in
order to determine whether or not to reject the null hypothesis.
Conversely, we can use the printed p-value to make the determination.
The procedures for handling directional hypotheses are identical.
187
Verification of the Assumption of Normality
Typically takes three forms:
th
188
Verification of the Assumption of Normality
Typically takes three forms:
th
1. Visual inspection of the distribution of residuals
189
Verification of the Assumption of Normality
Typically takes three forms:
th
1. Visual inspection of the distribution of residuals
2. Normal Probability Plot
190
Verification of the Assumption of Normality
Typically takes three forms:
th
1. Visual inspection of the distribution of residuals
2. Normal Probability Plot
3. Formal statistical test (Anderson-Darling, Kolmogorov-Smirnov,
Shapiro-Wilks)
191
Visual Inspection of the Residuals
Normality Assumption Met
192
Visual Inspection of the Residuals
Normality Assumption Met
193
Visual Inspection of the Residuals
Normality Assumption Not Met
194
Normal Probability Plot (Q-Q Plot) of the Residuals
Normality Assumption Met
195
Normal Probability Plot of the Residuals
Normality Assumption Not Met
196
Formal Statistical Test of Normality
Normality Assumption Met
197
Formal Statistical Test of Normality
Normality Assumption Met
198
Formal Statistical Test of Normality
Normality Assumption Met
Beware of using only a statistical test for normality: a tests ability to reject the null
hypothesis increases with the sample size; as the sample size becomes larger,
increasingly smaller departures from normality can be detected.
Because small departures from normality do not severely affect the validity of
hypothesis tests about slope and intercept coefficients, you should also examine plots
to make a final assessment of normality.
For small sample sizes, power is low for detecting even large departures from
normality that may be important. To increase the tests ability to detect such
deviations, you may want to consider using higher levels (such as =0.15 or 0.20)
rather than 0.05.
199
Formal Statistical Test of Normality
Normality Assumption Met
Beware of using only a statistical test for normality: a tests ability to reject the null
hypothesis increases with the sample size; as the sample size becomes larger,
increasingly smaller departures from normality can be detected.
Because small departures from normality do not severely affect the validity of
hypothesis tests about slope and intercept coefficients, you should also examine plots
to make a final assessment of normality.
For small sample sizes, power is low for detecting even large departures from
normality that may be important. To increase the tests ability to detect such
deviations, you may want to consider using higher levels (such as =0.15 or 0.20)
rather than 0.05.
200
Formal Statistical Test of Normality
Normality Assumption Met
Beware of using only a statistical test for normality: a tests ability to reject the null
hypothesis increases with the sample size; as the sample size becomes larger,
increasingly smaller departures from normality can be detected.
Because small departures from normality do not severely affect the validity of
hypothesis tests about slope and intercept coefficients, you should also examine plots
to make a final assessment of normality.
For small sample sizes, power is low for detecting even large departures from
normality that may be important. To increase the tests ability to detect such
deviations, you may want to consider using higher levels (such as =0.15 or 0.20)
rather than 0.05.
201
Formal Statistical Test of Normality
Normality Assumption Not Met
202
Stretch Question: What is the Null Hypothesis
Underlying the Formal Tests of Normality?
203
0 Residuals are not normally distributed in the population
1 Residuals are normally distributed in the population
NonNon-Response Grid
Stretch Question: What is the Null Hypothesis
Underlying the Formal Tests of Normality?
0 Residuals are not normally distributed in the population
1 Residuals are normally distributed in the population
204
0
0
NonNon-Response Grid
Stretch Question: What is the Null Hypothesis
Underlying the Formal Tests of Normality?
0 Residuals are not normally distributed in the population
1 Residuals are normally distributed in the population
205
0
0
Implication: We will never be able to conclude with reasonable
certainty that the residuals are normally distributed in
the population; we will only be able to conclude with
reasonable certainty that the residuals are not normally
di
distributed in the population.
th
NonNon-Response Grid
Stretch Question: What is the Null Hypothesis
Underlying the Formal Tests of Normality?
0 Residuals are not normally distributed in the population
1 Residuals are normally distributed in the population
206
0
0
Implication: We will never be able to conclude with reasonable
certainty that the residuals are normally distributed in
the population; we will only be able to conclude with
reasonable certainty that the residuals are not normally
di
distributed in the population.
th
Note, importantly, that failing to conclude that the
residuals are not normally distributed in the population
is not equivalent to concluding that they therefore are Non-Response Grid
Nonnormally distributed in the population.
Regression Through the Origin
Y=1X +
207
Regression Through the Origin
Y=1X +
1
( XY )
X
2
208
Regression Through the Origin
Y=1X +
1
( XY )
X
2
2
2
n 1
N.B.
209
210
Regression Through the Origin
Y=1X +
1
( XY )
X
2
2
2
n 1
N.B.
1
X
2
211
Regression Through the Origin
Y=1X +
1
( XY )
X
2
2
2
n 1
N.B.
Sum of the residuals need not be zero
1
X
2
212
Regression Through the Origin
Y=1X +
1
( XY )
X
2
2
2
n 1
N.B.
1
X
2
Sum of the residuals need not be zero
r2 can be negative and is not interpretable as a P.R.E. measure
213
Regression Through the Origin
Y=1X +
1
( XY )
X
2
2
2
n 1
N.B.
1
X
2
Sum of the residuals need not be zero
r2 can be negative and is not interpretable as a P.R.E. measure
Even if theoretically justifiable, BE CAREFUL!!!