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 n
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 avera
Statistical Method Final (2016-12-19)
Student ID # :
Name :
1.
(30 points) In an experiment to compare 3 brands of chocolate chip cookies (Pepperidge
Farms, SnackWell, and Chips Ahoy), thirty randomly
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
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 )
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 ),
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 fro
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
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
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
(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 eq
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 ,
(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% co
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 contai
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 contai
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 i
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,
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
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)
Rejecti
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 l
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 > MS
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
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
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 M
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
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 (
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
n
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: deter
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 o
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 j