Example 5
Dr. Sherri Brocks diet pills are supposed to cause significant weight loss.
The following table shows the results of a recent study where some
individuals took the diet pills and some did not.
Diet Pills
No Diet Pills
Total
No Weight Loss
80
20
Useful Calculation Formula for b
b=
S xx = n ( xi - x ) 2
i =1
= n xi ( xi - x )
i =1
= in=1 xi2 - n x 2
n
, where S xy = i=1 ( yi - y )( xi - x )
n
= i =1 yi ( xi - x )
n
= i =1 xi ( yi - y )
= in=1 xi yi - n x y
S xy
S xx
Note that each pair of
c = j =1 i =1
c
2
r
(X
ij
- E[ X ij ]
E[ X ]
)
2
= j =1 i =1
c
r
(X
ij
ij
- npi q j )
2
npi q j
Under H 0 ( Pij = pi q j , for all i, j ),
if n is large ( npi q j 5 , i = 1,., r and j = 1,., c ),
c = j =1 i=1
2
c
r
(X
ij
- npi q j )
2
~ c 2 ( r - 1
Goodness of Fit Tests when Some Parameters are Unspecified (Normal Case)
Let Y1 , Y2 ,., Yn be n independent samples from unknown distribution.
Test to determine whether
Y1 , Y2 ,., Yn were drawn from Normal
distribution.
Hypothesis
H 0 : Y is Normal di
D. Test of Independence in Contingency Tables
Many times, the n elements of a sample from a population may be classified
according to two different criteria. It is then of interest to know whether the
two methods of classification are statistically indepe
X 1 , X 2 ,., X k ~ M (n, p1 ,., pk ) and the sample size
( npi 5 , i = 1, 2,., k ),
If
is large
n
( X i - E[ X i ] ) 2
( X i - npi ) 2
k
i=1 E[ X ] = i =1 np ~ c 2 (k - 1) approximately.
i
i
k
C. A Goodness-of-Fit Test for the Multinomial Experiment
yij . : the mean of all observations in the (i , j ) th cell
n
yij . = k =1 yijk and yij . =
n
k =1
yijk
i = 1, 2,., a
j = 1, 2,., b
,
n
y. : the grand total of all observations
y. : the grand mean of all observations
a
b
n
y. = i =1 j =1 k =1 yijk an
Example 2
The dean of a business school was examining the factors that lead to
success in the MBA program. She felt that the type of degree and whether
the student had previous work experience were likely to be critical factors.
To test her beliefs she t
Test Statistic for Differences between the Treatments of Factor A
F=
MSA
SSA / ( a - 1)
=
~ F [ a - 1,(ab( n - 1)]
MSE SSE / ( ab( n - 1)
Under H 0 : t 1 = t 2 = . = t a = 0 , MSA MSE .
If MSA > MSE significantly, then reject H 0 . (Always Upper Tail)
Sampling Distribution of
b -b
S .E.[ b ]
b -b
~ Normal [ 0 ,1]
s2
s2
1) b ~ Normal [ b ,
]
S xx
S xx
2)
SSE
s
2
=
( n - 2)s 2
s
2
~ c 2 [ n - 2]
3) SSE is independent of b
b -b
s2
T=
S xx
(n - 2)s 2
=
( n - 2)
s2
b -b
b -b
=
~ t[ n - 2]
s2
S .E.[ b ]
S
(c) Compute the coefficient of determination.
(d) Interpret the meaning of the value of the coefficient of determination
that you found in Part c.
Be very specific.
(e) Use the estimated regression equation and predict sales for an
advertising expenditure
Test Statistic
F=
MSR
SSR / k
=
~ F [k - 1, n - k - 1]
MSE SSE / ( n - k - 1)
Under H 0 : b1 = b 2 = . = b k = 0 , MSR MSE .
If MSR > MSE significantly, then reject H 0 (Always Upper Tail)
Rejection Rule
Critical Value Approach
Reject H 0 if F =
p-val
Patterns for Residual Plots
Standardized Residuals and Outliers
We may also standardize the residuals by computing di =
ei
s2
, i = 1, 2,., n . If
the errors are normally distributed, approximately 95% of the standardized residuals
d i should fall in th
Example 3
Below you are given a partial computer output based on a sample of 12
observations relating the number of personal computers sold by a computer shop
per month ( Y ), unit price ( X 1 in $1,000) and the number of advertising spots ( X 2 )
used o
2) The t-test is used to determine whether each of the individual independent variables
is significant. A separate t-test is conducted for each of the independent variables in
the model. (a test for individual significance)
Use of t-Tests
For each indepe
Ch 8. Multiple Regression
A. Multiple Regression Model
Many applications of regression analysis involve situations in which there are more
than one independent variable. A regression model that contains more than one
independent variable is called a multi
Ch 8. Multiple Regression
A. Multiple Regression Model
Many applications of regression analysis involve situations in which there are more
than one independent variable. A regression model that contains more than one
independent variable is called a multi
(b) You want to test to see if there is a significant relationship between the
interest rate and monthly sales at the 1% level of significance.
State
the null and alternative hypotheses.
(c) At 99% confidence, test the hypotheses(Use the t test).
(d) Cons
Confidence Interval about the mean response E[ y0 ]
For a given value of x = x0 , there is a population of values of y0 whose mean is
E[ y0 ] = a + b x0 . To estimate the mean of y0 , given x = x0 , we would use the
following interval.
100(1 - a )% Conf
Example 2
A large catalogue chain store has been experimenting with several methods
of advertising its extensive variety of bicycles. Three kinds of catalogues
have been prepared. In one, a side view of each bicycle is shown. In another,
each bicycles ex
Comparing estimates of the population variance( s 2 )
1) Treatment Effect of Factor A
If the null hypothesis ( H 0 : t 1 = t 2 = . = t a = 0 ) is true,
the value of
MSA
1 . If not,
MSE
MSA
will be inflated because MSA overestimates s 2 .
MSE
2) Treatmen
Example 5
Three different brands of tires were compared for wear characteristics. For
each brand of tire, ten tires were randomly selected and subjected to
standard wear testing procedures.
The average mileage obtained for each
brand of tire and sample s
Sampling Distribution of the Test Statistic
T=
d - md
: t [ n - 1]
Sd / n
Under the Null Hypothesis ( H 0 : m d = 0 ),
T=
d
: t [ n - 1]
Sd / n
100(1 - a )% Confidence Interval on a Difference in Means
Sd
S
, d + ta / 2, n-1 d
d - ta / 2,n -1
n
n
S
Inference About A Population Mean Examples
Example 1
All cigarettes presently on the market have an average nicotine content of
at least 1.6 mg per cigarette. A firm that produces cigarettes claims that it
has discovered a new way to cure tobacco leaves
C. Concepts of Hypothesis Testing
Suppose that in order to test a specific null hypothesis H 0 , a random sample
of size n from the population , X 1 , X 2 ,., X n is to be observed.
Test Statistic ( T = T ( X 1 , X 2 ,., X n ) )
Calculated from the sampl
Summary of the Test
Rejection Rule
Hypothesis
Critical Value Approach
H0 : mX = mY
H 1 : m X > mY
H0 : mX = mY
H 1 : m X < mY
(X - Y )
Reject H 0 if Z =
Reject H 0 if Z =
2
sX
+
nX
2
sX
+
s Y2
2
sX
nX
+
< - za
Rejects H 0
if p-value < a
nY
(X -Y )
Reject
Review 3. Estimation
A. Statistical Inference
Statistical Inference is the process by which we acquire information about
populations from samples
Two types of Statistical Inference
Estimation: determining the value of a population parameter on the basis
Chi-Square Distribution
If Z1 , Z 2 ,., Z n are independent standard normal random variables, then X ,
defined by X = Z12 + Z 22 + L + Z n2 is said to have a chi-square Distribution with
n degrees of freedom ( X ~ c 2 [n] )
If X 1 and X 2 are independe
D. Probability Distribution for Two Random Variables
Bivariate Distribution: the rule for describing the probabilities of combinations of two
random variables
The Joint Probability Distribution
The joint probability mass function of two discrete random v
Sampling Distribution of the Test Statistic
2
SX
F=
2
sX
SY2
~ F [ n X - 1, nY - 1]
s Y2
100(1 - a )% Confidence Interval on a Difference in Variances (
2
2
SX
1
SX
, F
2
SY F /2, n -1,n -1 SY2 a /2,n -1,n
a
Y
X
Y
X
2
sX
)
s Y2
-1
2
2
Under the Nul