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Slides Prepared by
JOHN S. LOUCKS
St. Edward’s University
© 2003 South-Western/Thomson Learning™
Slide 1
Chapter 10
Statistical Inferences about Means
and Proportions for Two Populations




Estimation of the Difference Between the Means of
Two Populations: Independent Samples
Hypothesis Tests about the Difference Between the
Means of Two Populations: Independent Samples
Inferences about the Difference Between the Means
of Two Populations: Matched Samples
Inferences about the Difference Between the
Proportions of Two Populations
© 2003 South-Western/Thomson Learning™
Slide 2
Estimation of the Difference between the Means
of Two Populations: Independent Samples




Point Estimator of the Difference between the Means
of Two Populations
x1  x2
Sampling Distribution
Interval Estimate of Large-Sample Case
Interval Estimate of Small-Sample Case
© 2003 South-Western/Thomson Learning™
Slide 3
Point Estimator of the Difference between
the Means of Two Populations





Let 1 equal the mean of population 1 and 2 equal
the mean of population 2.
The difference between the two population means is
1 - 2.
To estimate 1 - 2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2.
Let x1 equal the mean of sample 1 and x2 equal the
mean of sample 2.
The point estimator of the difference between the
means of the populations 1 and 2 is x1  x2 .
© 2003 South-Western/Thomson Learning™
Slide 4
Sampling Distribution of x1  x2

Properties of the Sampling Distribution of x1  x2
• Expected Value
E ( x1  x2 )  1   2
• Standard Deviation
 x1  x2 
12
n1

 22
n2
where: 1 = standard deviation of population 1
2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2
© 2003 South-Western/Thomson Learning™
Slide 5
Interval Estimate of 1 - 2:
Large-Sample Case (n1 > 30 and n2 > 30)

Interval Estimate with 1 and 2 Assumed Known
where:

x1  x2  z / 2  x1  x2
1 -  is the confidence coefficient
Interval Estimate with 1 and 2 Estimated by s1 and s2
x1  x2  z / 2 sx1  x2
where:
sx1  x2
© 2003 South-Western/Thomson Learning™
s12 s22


n1 n2
Slide 6
Example: Par, Inc.

Interval Estimate of 1 - 2: Large-Sample Case
Par, Inc. is a manufacturer of golf equipment and
has developed a new golf ball that has been designed
to provide “extra distance.” In a test of driving
distance using a mechanical driving device, a sample of
Par golf balls was compared with a sample of golf balls
made by Rap, Ltd., a competitor.
The sample statistics appear on the next slide.
© 2003 South-Western/Thomson Learning™
Slide 7
Example: Par, Inc.

Interval Estimate of 1 - 2: Large-Sample Case
• Sample Statistics
Sample Size
Mean
Standard Dev.
Sample #1
Par, Inc.
n1 = 120 balls
x1 = 235 yards
s1 = 15 yards
© 2003 South-Western/Thomson Learning™
Sample #2
Rap, Ltd.
n2 = 80 balls
x2 = 218 yards
s2 = 20 yards
Slide 8
Example: Par, Inc.

Point Estimate of the Difference Between Two
Population Means
1 = mean distance for the population of
Par, Inc. golf balls
2 = mean distance for the population of
Rap, Ltd. golf balls
Point estimate of 1 - 2 = x1  x2 = 235 - 218 = 17 yards.
© 2003 South-Western/Thomson Learning™
Slide 9
Point Estimator of the Difference
between the Means of Two Populations
Population 1
Par, Inc. Golf Balls
Population 2
Rap, Ltd. Golf Balls
1 = mean driving
2 = mean driving
distance of Par
golf balls
distance of Rap
golf balls
m1 – 2 = difference between
the mean distances
Simple random sample
of n1 Par golf balls
Simple random sample
of n2 Rap golf balls
x1 = sample mean distance
for sample of Par golf ball
x2 = sample mean distance
for sample of Rap golf ball
x1 - x2 = Point Estimate of 1 – 2
© 2003 South-Western/Thomson Learning™
Slide 10
Example: Par, Inc.

95% Confidence Interval Estimate of the Difference
Between Two Population Means: Large-Sample Case,
1 and 2 Estimated by s1 and s2
Substituting the sample standard deviations for
the population standard deviation:
x1  x2  z / 2
12
 22
(15) 2 ( 20) 2

 17  1. 96

n1 n2
120
80
= 17 + 5.14 or 11.86 yards to 22.14 yards.
We are 95% confident that the difference between the
mean driving distances of Par, Inc. balls and Rap, Ltd.
balls lies in the interval of 11.86 to 22.14 yards.
© 2003 South-Western/Thomson Learning™
Slide 11
Using Excel to Develop an Interval Estimate
of 1 – 2: Large-Sample Case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Formula Worksheet
A
Par
195
230
254
205
260
222
241
217
228
255
209
251
229
220
B
C
Rap
226 Sample Size
198
Mean
203
Stand. Dev.
237
235 Confid. Coeff.
204 Lev. of Signif.
199
z Value
202
240
Std. Error
221 Marg. of Error
206
201 Pt. Est. of Diff.
233
Lower Limit
194
Upper Limit
D
Par, Inc.
120
=AVERAGE(A2:A121)
=STDEV(A2:A121)
E
Rap, Ltd.
80
=AVERAGE(A2:A81)
=STDEV(A2:A81)
0.95
=1-D6
=NORMSINV(1-D7/2)
=SQRT(D4^2*/D2+E4^2/E2)
=D8*D10
=D3-E3
=D13-D11
=D13+D11
Note: Rows 16-121 are not shown.
© 2003 South-Western/Thomson Learning™
Slide 12
Using Excel to Develop an Interval Estimate
of 1 – 2: Large-Sample Case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Value Worksheet
A
Par
195
230
254
205
260
222
241
217
228
255
209
251
229
220
B
C
Rap
226 Sample Size
198
Mean
203
Stand. Dev.
237
235 Confid. Coeff.
204 Lev. of Signif.
199
z Value
202
240
Std. Error
221 Marg. of Error
206
201 Pt. Est. of Diff.
233
Lower Limit
194
Upper Limit
D
Par, Inc.
120
235
15
E
Rap, Ltd.
80
218
20
0.95
0.05
1.960
2.622
5.139
17
11.86
22.14
Note: Rows 16-121 are not shown.
© 2003 South-Western/Thomson Learning™
Slide 13
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)

Interval Estimate with  2 Assumed Known
x1  x2  z / 2  x1  x2
where:
 x1  x2
© 2003 South-Western/Thomson Learning™
1 1
  (  )
n1 n2
2
Slide 14
Interval Estimate of 1 - 2:
Small-Sample Case (n1 < 30 and/or n2 < 30)

Interval Estimate with 1 and 2 Estimated by s1 and s2
x1  x2  t / 2 sx1  x2
where:
sx1  x2
1 1
 s (  )
n1 n2
2
© 2003 South-Western/Thomson Learning™
2
2
(
n

1
)
s

(
n

1
)
s
1
2
2
s2  1
n1  n2  2
Slide 15
Example: Specific Motors
Specific Motors of Detroit has developed a new
automobile known as the M car. 12 M cars and 8 J cars
(from Japan) were road tested to compare miles-pergallon (mpg) performance. The sample statistics are:
Sample Size
Mean
Standard Deviation
© 2003 South-Western/Thomson Learning™
Sample #1
M Cars
n1 = 12 cars
x1 = 29.8 mpg
s1 = 2.56 mpg
Sample #2
J Cars
n2 = 8 cars
x2 = 27.3 mpg
s2 = 1.81 mpg
Slide 16
Example: Specific Motors

Point Estimate of the Difference Between Two
Population Means
1 = mean miles-per-gallon for the population of
M cars
2 = mean miles-per-gallon for the population of
J cars
Point estimate of 1 - 2 = x1  x2 = 29.8 - 27.3 = 2.5
mpg.
© 2003 South-Western/Thomson Learning™
Slide 17
Example: Specific Motors

95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
We will make the following assumptions:
• The miles per gallon rating must be normally
distributed for both the M car and the J car.
• The variance in the miles per gallon rating must
be the same for both the M car and the J car.
Using the t distribution with n1 + n2 - 2 = 18 degrees
of freedom, the appropriate t value is t.025 = 2.101.
We will use a weighted average of the two sample
variances as the pooled estimator of  2.
© 2003 South-Western/Thomson Learning™
Slide 18
Example: Specific Motors

95% Confidence Interval Estimate of the Difference
Between Two Population Means: Small-Sample Case
2
2
2
2
(
n

1
)
s

(
n

1
)
s
11
(
2
.
56
)

7
(
1
.
81
)
1
2
2
s2  1

 5. 28
n1  n2  2
12  8  2
x1  x2  t.025
1 1
1 1
s (  )  2. 5  2.101 5. 28(  )
n1 n2
12 8
2
= 2.5 + 2.2 or .3 to 4.7 miles per gallon.
We are 95% confident that the difference between the
mean mpg ratings of the two car types is from 0.3 to
4.7 mpg (with the M car having the higher mpg).
© 2003 South-Western/Thomson Learning™
Slide 19
Using Excel to Develop an Interval Estimate
of 1 – 2: Small-Sample Case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Formula Worksheet
A
M Car
25.1
32.2
31.7
27.6
28.5
33.6
30.8
26.2
29.0
31.0
31.7
30.0
B
C
D
J Car
M Car
25.6 Sample Size 12
28.1
Mean =AVERAGE(A2:A13)
27.9
Stand. Dev. =STDEV(A2:A13)
25.3
30.1 Confid. Coeff. 0.95
27.5 Lev. of Signif. =1-D6
25.1
Deg. Freed. =D2+E2-2
28.8
z Value =TINV(D7,D8)
E
J Car
8
=AVERAGE(B2:B9)
=STDEV(B2:B9)
Pool.Est.Var. =((D2-1)*D4^2+(E2-1)*E4^2)/D8
Std. Error =SQRT(D11*(1/D2+1/E2))
Marg. of Error =D9*D12
Pt. Est. of Diff. =D3-E3
Lower Limit =D15-D13
Upper Limit =D15+D13
© 2003 South-Western/Thomson Learning™
Slide 20
Using Excel to Develop an Interval Estimate
of 1 – 2: Small-Sample Case

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
A
M Car
25.1
32.2
31.7
27.6
28.5
33.6
30.8
26.2
29.0
31.0
31.7
30.0
B
C
J Car
25.6 Sample Size 12
28.1
Mean 29.8
27.9
Stand. Dev. 2.56
25.3
30.1 Confid. Coeff. 0.95
27.5 Lev. of Signif. 0.05
25.1
Deg. Freed. 18
28.8
z Value 2.101
D
M Car
E
J Car
8
27.3
1.81
Pool.Est.Var. 5.2765
Std. Error 1.0485
Marg. of Error 2.2027
Pt. Est. of Diff. 2.4833
Lower Limit 0.2806
Upper Limit 4.6861
© 2003 South-Western/Thomson Learning™
Slide 21
Hypothesis Tests about the Difference
between the Means of Two Populations:
Independent Samples

Hypotheses
H0: 1 - 2 < 0
Ha: 1 - 2 > 0

H0: 1 - 2 > 0
Ha: 1 - 2 < 0
Test Statistic
Large-Sample
z
( x1  x2 )  ( 1   2 )
12 n1   22 n2
© 2003 South-Western/Thomson Learning™
H0: 1 - 2 = 0
Ha: 1 - 2  0
Small-Sample
t
( x1  x2 )  ( 1   2 )
s2 (1 n1  1 n2 )
Slide 22
Example: Par, Inc.

Hypothesis Tests About the Difference Between the
Means of Two Populations: Large-Sample Case
Par, Inc. is a manufacturer of golf equipment and has
developed a new golf ball that has been designed to
provide “extra distance.” In a test of driving distance
using a mechanical driving device, a sample of Par
golf balls was compared with a sample of golf balls
made by Rap, Ltd., a competitor. The sample
statistics appear on the next slide.
© 2003 South-Western/Thomson Learning™
Slide 23
Example: Par, Inc.

Hypothesis Tests about the Difference between the
Means of Two Populations: Large-Sample Case
• Sample Statistics
Sample Size
Mean
Standard Dev.
Sample #1
Par, Inc.
n1 = 120 balls
x1 = 235 yards
s1 = 15 yards
© 2003 South-Western/Thomson Learning™
Sample #2
Rap, Ltd.
n2 = 80 balls
x2 = 218 yards
s2 = 20 yards
Slide 24
Example: Par, Inc.

Hypothesis Tests about the Difference between the
Means of Two Populations: Large-Sample Case
Can we conclude, using a .01 level of
significance, that the mean driving distance of Par,
Inc. golf balls is greater than the mean driving
distance of Rap, Ltd. golf balls?
1 = mean distance for the population of Par, Inc.
golf balls
2 = mean distance for the population of Rap, Ltd.
golf balls
• Hypotheses H0: 1 - 2 < 0
Ha: 1 - 2 > 0
© 2003 South-Western/Thomson Learning™
Slide 25
Example: Par, Inc.

Hypothesis Tests about the Difference between the
Means of Two Populations: Large-Sample Case
• Rejection Rule
Reject H0 if z > 2.33
z
( x1  x2 )  ( 1   2 )
12
n1

 22
n2
( 235  218)  0
17


 6. 49
2
2
2. 62
(15) ( 20)

120
80
• Conclusion
Reject H0. We are at least 99% confident
that the mean driving distance of Par, Inc. golf balls is
greater than the mean driving distance of Rap, Ltd.
golf balls.
© 2003 South-Western/Thomson Learning™
Slide 26
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Large Sample Case

Excel’s “z-Test: Two Sample for Means” Tool
Step 1 Select the Tools pull-down menu
Step 2 Choose the Data Analysis option
Step 3 Choose z-Test: Two Sample for Means
from the list of Analysis Tools
… continued
© 2003 South-Western/Thomson Learning™
Slide 27
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Large Sample Case

Excel’s “z-Test: Two Sample for Means” Tool
Step 4 When the z-Test: Two Sample for Means
dialog box appears:
Enter A1:A121 in the Variable 1 Range box
Enter B1:B81 in the Variable 2 Range box
Enter 0 in the Hypothesized Mean Difference
box
Enter 225 in the Variable 1 Variance (known)
box
Enter 400 in the Variable 2 Variance (known)
box
… continued
© 2003 South-Western/Thomson Learning™
Slide 28
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Large Sample Case

Excel’s “z-Test: Two Sample for Means” Tool
Step 4 (continued)
Select Labels
Enter .01 in the Alpha box
Select Output Range
Enter D4 in the Output Range box
(Any upper left-hand corner cell indicating
where the output is to begin may be entered)
Click OK
© 2003 South-Western/Thomson Learning™
Slide 29
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Large Sample Case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Value Worksheet
A
Par
195
230
254
205
260
222
241
217
228
255
209
251
229
220
B C
D
Rap
226
Sample Variance
198
203
z-Test: Two Sample for Means
237
235
204
Mean
199
Known Variance
202
Observations
240
Hypothesized Mean Difference
221
z
206
P(Z<=z) one-tail
201
z Critical one-tail
233
P(Z<=z) two-tail
194
z Critical two-tail
© 2003 South-Western/Thomson Learning™
E
Par, Inc.
225
F
Rap, Ltd.
400
Par, Inc.
Rap, Ltd.
235
218
225
400
120
80
0
6.483545607
4.50145E-11
2.326341928
9.00291E-11
2.575834515
Note: Rows 16-121 are not shown.
Slide 30
Example: Specific Motors

Hypothesis Tests about the Difference between the
Means of Two Populations: Small-Sample Case
Can we conclude, using a .05 level of
significance, that the miles-per-gallon (mpg)
performance of M cars is greater than the miles-pergallon performance of J cars?
1 = mean mpg for the population of M cars
2 = mean mpg for the population of J cars
• Hypotheses H0: 1 - 2 < 0
Ha: 1 - 2 > 0
© 2003 South-Western/Thomson Learning™
Slide 31
Example: Specific Motors

Hypothesis Tests about the Difference between the
Means of Two Populations: Small-Sample Case
• Rejection Rule
Reject H0 if t > 1.734
(a = .05, d.f. = 18)
• Test Statistic
t
( x1  x2 )  ( 1   2 )
s2 (1 n1  1 n2 )
where:
2
2
(
n

1)
s

(
n

1)
s
1
2
2
s2  1
n1  n2  2
© 2003 South-Western/Thomson Learning™
Slide 32
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Small Sample Case

Excel’s “t-Test: Two Sample Assuming Equal
Variances” Tool
Step 1 Select the Tools pull-down menu
Step 2 Choose the Data Analysis option
Step 3 Choose t-Test: Two Sample Assuming Equal
Variances from the list of Analysis Tools
… continued
© 2003 South-Western/Thomson Learning™
Slide 33
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Small Sample Case

Excel’s “t-Test: Two Sample Assuming Equal
Variances” Tool
Step 4 When the t-Test: Two Sample Assuming
Equal Variances dialog box appears:
Enter A1:A13 in the Variable 1 Range box
Enter B1:B9 in the Variable 2 Range box
Enter 0 in the Hypothesized Mean
Difference box
… continued
© 2003 South-Western/Thomson Learning™
Slide 34
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Small Sample Case

Excel’s “t-Test: Two Sample Assuming Equal
Variances” Tool
Step 4 (continued)
Select Labels
Enter .01 in the Alpha box
Select Output Range
Enter D1 in the Output Range box
(Any upper left-hand corner cell indicating
where the output is to begin may be entered)
Click OK
© 2003 South-Western/Thomson Learning™
Slide 35
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Small Sample Case

1
2
3
4
5
6
7
8
9
10
11
12
13
14
Value Worksheet
A
M Car
25.1
32.2
31.7
27.6
28.5
33.6
30.8
26.2
29.0
31.0
31.7
30.0
B
J Car
25.6
28.1
27.9
25.3
30.1
27.5
25.1
28.8
C
D
E
F
t-Test: Two-Sample Assuming Equal Variances
M Car
J Car
Mean
29.78333
27.3
Variance
6.556061
3.265714
12
8
Observations
Pooled Variance
Hypothesized Mean Diff.
df
5.276481
0
18
t Stat
2.368555
P(T<=t) one-tail
0.014626
t Critical one-tail
1.734063
P(T<=t) two-tail
0.029251
t Critical two-tail
2.100924
© 2003 South-Western/Thomson Learning™
Slide 36
Inference about the Difference between the
Means of Two Populations: Matched Samples



With a matched-sample design each sampled item
provides a pair of data values.
The matched-sample design can be referred to as
blocking.
This design often leads to a smaller sampling error
than the independent-sample design because
variation between sampled items is eliminated as a
source of sampling error.
© 2003 South-Western/Thomson Learning™
Slide 37
Example: Express Deliveries

Inference about the Difference between the Means of
Two Populations: Matched Samples
A Chicago-based firm has documents that must
be quickly distributed to district offices throughout
the U.S. The firm must decide between two delivery
services, UPX (United Parcel Express) and INTEX
(International Express), to transport its documents.
In testing the delivery times of the two services, the
firm sent two reports to a random sample of ten
district offices with one report carried by UPX and
the other report carried by INTEX.
Do the data that follow indicate a difference in
mean delivery times for the two services?
© 2003 South-Western/Thomson Learning™
Slide 38
Example: Express Deliveries
District Office
Seattle
Los Angeles
Boston
Cleveland
New York
Houston
Atlanta
St. Louis
Milwaukee
Denver
Delivery Time (Hours)
UPX
INTEX
Difference
32
30
19
16
15
18
14
10
7
16
© 2003 South-Western/Thomson Learning™
25
24
15
15
13
15
15
8
9
11
7
6
4
1
2
3
-1
2
-2
5
Slide 39
Example: Express Deliveries

Inference about the Difference between the Means of
Two Populations: Matched Samples
Let d = the mean of the difference values for the
two delivery services for the population of
district offices
• Hypotheses
• Rejection Rule
H0: d = 0, Ha: d 
Assuming the population of difference values is
approximately normally distributed, the t
distribution with n - 1 degrees of freedom applies.
With  = .05, t.025 = 2.262 (9 degrees of freedom).
Reject H0 if t < -2.262 or if t > 2.262
© 2003 South-Western/Thomson Learning™
Slide 40
Example: Express Deliveries

Inference about the Difference between the Means of
Two Populations: Matched Samples
 di ( 7  6... 5)
d 

 2. 7
n
10
2
76.1
 ( di  d )
sd 

 2. 9
n 1
9
d  d
2. 7  0
t

 2. 94
sd n 2. 9 10
• Conclusion
Reject H0.
There is a significant difference between the mean
delivery times for the two services.
© 2003 South-Western/Thomson Learning™
Slide 41
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Matched Samples

Excel’s “t-Test: Paired Two Sample for Means” Tool
Step 1 Select the Tools pull-down menu
Step 2 Choose the Data Analysis option
Step 3 Choose t-Test: Paired Two Sample for Means
from the list of Analysis Tools
… continued
© 2003 South-Western/Thomson Learning™
Slide 42
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Matched Samples

Excel’s “t-Test: Paired Two Sample for Means” Tool
Step 4 When the t-Test: Paired Two Sample for Means
dialog box appears:
Enter B1:B11 in the Variable 1 Range box
Enter C1:C11 in the Variable 2 Range box
Enter 0 in the Hypothesized Mean Difference
box
Select Labels
Enter .05 in the Alpha box
Select Output Range
Enter E2 (your choice) in the Output Range box
Click OK
© 2003 South-Western/Thomson Learning™
Slide 43
Using Excel to Conduct a Hypothesis Test
about 1 – 2: Matched Samples

Value Worksheet
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
A
B
C
D
E
F
G
Office
UPX INTEX
Seattle
32
25
t-Test: Paired Two Sample for Means
L.A.
30
24
Boston
19
15
UPX INTEX
Cleveland
16
15
Mean
17.7
15
N.Y.C.
15
13
Variance
62.011 31.7778
Houston
18
15
Observations
10
10
Atlanta
14
15
Pearson Correlation
0.9612
St. Louis
10
8
Hypothesized Mean Difference
0
Milwauk.
7
9
df
9
Denver
16
11
t Stat
2.9362
P(T<=t) one-tail
0.0083
t Critical one-tail
1.8331
P(T<=t) two-tail
0.0166
t Critical two-tail
2.2622
© 2003 South-Western/Thomson Learning™
Slide 44
Inferences about the Difference between the
Proportions of Two Populations



Sampling Distribution of p1  p2
Interval Estimation of p1 - p2
Hypothesis Tests about p1 - p2
© 2003 South-Western/Thomson Learning™
Slide 45
Sampling Distribution of p1  p2

Expected Value
E ( p1  p2 )  p1  p2

Standard Deviation
 p1  p2 

p1 (1  p1 ) p2 (1  p2 )

n1
n2
Distribution Form
If the sample sizes are large (n1p1, n1(1 - p1), n2p2,
and n2(1 - p2) are all greater than or equal to 5), the
sampling distribution of p1  p2 can be approximated
by a normal probability distribution.
© 2003 South-Western/Thomson Learning™
Slide 46
Interval Estimation of p1 - p2

Interval Estimate
p1  p2  z / 2  p1  p2

Point Estimator of  p1  p2
s p1  p2 
p1 (1  p1 ) p2 (1  p2 )

n1
n2
© 2003 South-Western/Thomson Learning™
Slide 47
Example: MRA
MRA (Market Research Associates) is conducting
research to evaluate the effectiveness of a client’s new
advertising campaign. Before the new campaign
began, a telephone survey of 150 households in the
test market area showed 60 households “aware” of the
client’s product. The new campaign has been initiated
with TV and newspaper advertisements running for
three weeks. A survey conducted immediately after
the new campaign showed 120 of 250 households
“aware” of the client’s product.
Does the data support the position that the
advertising campaign has provided an increased
awareness of the client’s product?
© 2003 South-Western/Thomson Learning™
Slide 48
Example: MRA

Point Estimator of the Difference between the
Proportions of Two Populations
120 60
p1  p2  p1  p2 

. 48. 40 . 08
250 150
p1 = proportion of the population of households
“aware” of the product after the new campaign
p2 = proportion of the population of households
“aware” of the product before the new
campaign
p1 = sample proportion of households “aware” of the
product after the new campaign
p2 = sample proportion of households “aware” of the
product before the new campaign
© 2003 South-Western/Thomson Learning™
Slide 49
Example: MRA

Interval Estimate of p1 - p2: Large-Sample Case
For = .05, z.025 = 1.96:
. 48(.52) . 40(. 60)
. 48. 40  1. 96

250
150
.08 + 1.96(.0510)
.08 + .10
or -.02 to +.18
• Conclusion
At a 95% confidence level, the interval estimate of
the difference between the proportion of households
aware of the client’s product before and after the new
advertising campaign is -.02 to +.18.
© 2003 South-Western/Thomson Learning™
Slide 50
Using Excel to Develop
an Interval Estimate of p1 – p2

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Formula Worksheet
A
Sur2
No
Yes
Yes
No
Yes
No
No
Yes
No
Yes
Yes
Yes
No
Yes
B
Sur1
Yes
No
Yes
Yes
No
No
Yes
No
No
Yes
No
Yes
Yes
Yes
C
D
E
Survey 2 (from Popul.1) Survey 1 (from Popul.2)
Sample Size 250
150
No. of "Yes" =COUNTIF(A2:A251,"Yes") =COUNTIF(B2:B151,"Yes")
Samp. Propor. =D3/D2
=E3/E2
Confid. Coeff. 0.95
Lev. Of Signif. =1-D6
z Value =NORMSINV(1-D7/2)
Std. Error =SQRT(D4*(1-D4)/D2+E4*(1-E4)/E2)
Marg. of Error =D8*D10
Pt. Est. of Diff. =D4-E4
Lower Limit =D13-D11
Upper Limit =D13+D11
Note: Rows 16-251 are not shown.
© 2003 South-Western/Thomson Learning™
Slide 51
Using Excel to Develop
an Interval Estimate of p1 – p2

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Value Worksheet
A
Sur2
No
Yes
Yes
No
Yes
No
No
Yes
No
Yes
Yes
Yes
No
Yes
B
Sur1
Yes
No
Yes
Yes
No
No
Yes
No
No
Yes
No
Yes
Yes
Yes
C
D
E
Survey 2 (from Popul.1) Survey 1 (from Popul.2)
Sample Size 250
150
No. of "Yes" 120
60
Samp. Propor. 0.48
0.40
Confid. Coeff. 0.95
Lev. Of Signif. 0.05
z Value 1.960
Std. Error 0.0510
Marg. of Error 0.0999
Pt. Est. of Diff. 0.080
Lower Limit -0.020
Upper Limit 0.180
Note: Rows 16-251 are not shown.
© 2003 South-Western/Thomson Learning™
Slide 52
Hypothesis Tests about p1 - p2

Hypotheses
H0: p1 - p2 < 0
Ha: p1 - p2 > 0

Test statistic
z

( p1  p2 )  ( p1  p2 )
 p1  p2
Point Estimator of  p1  p2 where p1 = p2
s p1  p2  p (1  p )(1 n1  1 n2 )
where:
n1 p1  n2 p2
p
n1  n2
© 2003 South-Western/Thomson Learning™
Slide 53
Example: MRA

Hypothesis Tests about p1 - p2
Can we conclude, using a .05 level of
significance, that the proportion of households aware
of the client’s product increased after the new
advertising campaign?
p1 = proportion of the population of households
“aware” of the product after the new campaign
p2 = proportion of the population of households
“aware” of the product before the new
campaign
• Hypotheses
H0: p1 - p2 < 0
Ha: p1 - p2 > 0
© 2003 South-Western/Thomson Learning™
Slide 54
Example: MRA

Hypothesis Tests about p1 - p2
• Rejection Rule
Reject H0 if z > 1.645
• Test Statistic
250(. 48)  150(. 40) 180
p

. 45
250  150
400
s p1  p2  . 45(. 55)( 1
 1 ) . 0514
250 150
(. 48. 40)  0
. 08
z

 1. 56
. 0514
. 0514
• Conclusion
© 2003 South-Western/Thomson Learning™
Do not reject H0.
Slide 55
Using Excel to Conduct
a Hypothesis Test about p1 – p2

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Formula Worksheet
A
Sur2
No
Yes
Yes
No
Yes
No
No
Yes
No
Yes
Yes
Yes
No
Yes
Yes
B
C
Sur1
Yes
Sample Size
No
No. of "Yes"
Yes
Samp. Propor.
Yes
No
Lev of Signif.
No Crit.Val. (upper)
Yes
No
Pt. Est. of Diff.
No
Hypoth. Value
Yes
No
Pool. Est. of p
Yes
Standard Error
Yes
Test Statistic
Yes
p -Value
No
Conclusion
D
E
Survey 2 (from Popul.1) Survey 1 (from Popul.2)
250
150
=COUNTIF(A2:A251,"Yes") =COUNTIF(B2:B151,"Yes")
=D3/D2
=E3/E2
0.05
=NORMSINV(1-D7)
=D4-E4
0
=(D2*D4+E2*E4)/(D2+E2)
=SQRT(D12*(1-D12)*(1/D2+1/E2))
=(D9-D10)/D13
=2*NORMSDIST(D14)
=IF(D15<D6,"Reject","Do Not Reject")
© 2003 South-Western/Thomson Learning™
Note: Rows 17-251 are not shown.
Slide 56
Using Excel to Conduct
a Hypothesis Test about p1 – p2

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Value Worksheet
A
Sur2
No
Yes
Yes
No
Yes
No
No
Yes
No
Yes
Yes
Yes
No
Yes
Yes
B
C
Sur1
Yes
Sample Size
No
No. of "Yes"
Yes
Samp. Propor.
Yes
No
Lev of Signif.
No Crit.Val. (upper)
Yes
No
Pt. Est. of Diff.
No
Hypoth. Value
Yes
No
Pool. Est. of p
Yes
Standard Error
Yes
Test Statistic
Yes
p -Value
No
Conclusion
D
E
Survey 2 (from Popul.1) Survey 1 (from Popul.2)
250
150
120
60
0.48
0.40
0.05
1.645
0.08
0
0.450
0.0514
1.557
0.060
Do Not Reject
© 2003 South-Western/Thomson Learning™
Note: Rows 17-251 are not shown.
Slide 57
End of Chapter 10
© 2003 South-Western/Thomson Learning™
Slide 58