Download INFERENCE: TWO POPULATIONS

Slides by
John
Loucks
St. Edward’s
University
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
1
Chapter 10, Part A: Comparisons Involving Means,
Experimental Design, and Analysis of Variance

Inferences About the Difference Between
Two Population Means: s 1 and s 2 Known

Inferences About the Difference Between
Two Population Means: s 1 and s 2 Unknown

Inferences About the Difference Between
Two Population Means: Matched Samples
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
2
Inferences About the Difference Between
Two Population Means: s 1 and s 2 Known


Interval Estimation of m 1 – m 2
Hypothesis Tests About m 1 – m 2
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
3
Estimating the Difference Between
Two Population Means




Let m1 equal the mean of population 1 and m2 equal
the mean of population 2.
The difference between the two population means is
m1 - m2.
To estimate m1 - m2, 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 .
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
4
Sampling Distribution of 𝑥1 − 𝑥2

Expected Value
𝐸 𝑥1 − 𝑥2 = 𝜇1 − 𝜇2

Standard Deviation (Standard Error)
𝜎𝑥1 −𝑥2 =
𝜎1 2 𝜎2 2
+
𝑛1
𝑛2
where: s1 = standard deviation of population 1
s2 = standard deviation of population 2
n1 = sample size from population 1
n2 = sample size from population 2
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
5
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Interval Estimate
𝑥1 − 𝑥2 ± 𝑧𝛼/2
𝜎1 2 𝜎2 2
+
𝑛1
𝑛2
where:
1 -  is the confidence coefficient
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
6
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Example: Par, Inc.
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.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
7
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Example: Par, Inc.
Sample Size
Sample Mean
Sample #1
Par, Inc.
120 balls
275 yards
Sample #2
Rap, Ltd.
80 balls
258 yards
Based on data from previous driving distance
tests, the two population standard deviations are
known with s 1 = 15 yards and s 2 = 20 yards.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
8
Interval Estimation of m1 - m2:
s 1 and s 2 Known

Example: Par, Inc.
Let us develop a 95% confidence interval estimate
of the difference between the mean driving distances of
the two brands of golf ball.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
9
Estimating the Difference Between
Two Population Means
Population 1
Par, Inc. Golf Balls
m1 = mean driving
distance of Par
golf balls
Population 2
Rap, Ltd. Golf Balls
m2 = mean driving
distance of Rap
golf balls
m1 – m2 = 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 the Par golf balls
𝑥2 = sample mean distance
for the Rap golf balls
𝑥1 − 𝑥2 = Point Estimate of m1 – m2
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
10
Point Estimate of m1 - m2
Point estimate of m1 - m2 = 𝑥1 − 𝑥2
= 295 - 278
= 17 yards
where:
m1 = mean distance for the population
of Par, Inc. golf balls
m2 = mean distance for the population
of Rap, Ltd. golf balls
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
11
Interval Estimation of m1 - m2:
s 1 and s 2 Known
𝑥1 − 𝑥2 ± 𝑧𝛼/2
𝜎1 2 𝜎2 2
(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 is 11.86 to 22.14 yards.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
12
Interval Estimation of m1 - m2:
s 1 and s 2 Known

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Excel Formula Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291
289
282
B
C
D
E
Rap
Par, Inc.
Rap, Ltd.
266
Sample Size =COUNT(A2:A121)
=COUNT(B2:B81)
238
Sample Mean =AVERAGE(A2:A121)
=AVERAGE(B2:B81)
243
277 Popul. Std. Dev. 15
20
275 Standard Error =SQRT(D5^2/D2+E5^2/E2)
244
239
Confid. Coeff. 0.95
Note: Rows
242 Level of Signif. =1-D8
280
z Value =NORM.S.INV(1-D9/2)
16-121 are
261 Margin of Error =D10*D6
not shown.
276
241
Pt. Est. of Diff. =D3-E3
273
Lower Limit =D13-D11
248
Upper Limit =D13+D11
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
13
Interval Estimation of m1 - m2:
s 1 and s 2 Known

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Excel Value Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291
289
282
C
B
Rap
Sample Size 120
266
Sample Mean 275
238
243
277 Popul. Std. Dev. 15
275 Standard Error 2.622
244
Confid. Coeff. 0.95
239
242 Level of Signif. 0.05
z Value 1.96
280
261 Margin of Error 5.14
276
Pt. Est. of Diff. 17
241
Lower Limit 11.86
273
Upper Limit 22.14
248
E
Rap, Ltd.
D
Par, Inc.
80
258
20
Note: Rows
16-121 are
not shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
14
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Known
 Hypotheses
H0: m1 – m2 > D0 H0: m1 – m2 < D0 H0: m1 – m2 = D0
Ha: m1 – m2 < D0 Ha: m1 – m2 > D0 Ha: m1 – m2 ≠ D0
Left-tailed
Right-tailed
Two-tailed
 Test Statistic
𝑧=
𝑥1 − 𝑥2 − 𝐷0
𝜎1 2 𝜎2 2
+
𝑛1
𝑛2
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
15
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Known

Example: Par, Inc.
Can we conclude, using  = .01, that the mean
driving distance of Par, Inc. golf balls is greater
than the mean driving distance of Rap, Ltd. golf
balls?
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
16
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Known
 p-Value and Critical Value Approaches
1. Develop the hypotheses.
H0: m1 - m2 < 0
Ha: m1 - m2 > 0
Righttailed
test
where:
m1 = mean distance for the population
of Par, Inc. golf balls
m2 = mean distance for the population
of Rap, Ltd. golf balls
2. Specify the level of significance.
 = .01
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
17
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Known
 p-Value and Critical Value Approaches
3. Compute the value of the test statistic.
𝑧=
𝑧=
𝑥1 − 𝑥2 − 𝐷0
𝜎1 2 𝜎2 2
+
𝑛1
𝑛2
235 − 218 − 0
17
=
= 6.49
2.62
(15)2 (20)2
+
120
80
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
18
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Known
 p-Value Approach
4. Compute the p-value.
For z = 6.49, the p-value < .0001.
5. Determine whether to reject H0.
Because p–value <  = .01, we reject H0.
At the .01 level of significance, the sample evidence
indicates the mean driving distance of Par, Inc. golf
balls is greater than the mean driving distance of Rap,
Ltd. golf balls.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
19
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Known
 Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .01, z.01 = 2.33
Reject H0 if z > 2.33
5. Determine whether to reject H0.
Because z = 6.49 > 2.33, we reject H0.
The sample evidence indicates the mean driving
distance of Par, Inc. golf balls is greater than the mean
driving distance of Rap, Ltd. golf balls.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
20
Excel’s “z-Test: Two Sample for Means” Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose z-Test: Two Sample for Means
from the list of Analysis Tools
Step 4 When the z-Test: Two Sample for Means
dialog box appears:
(see details on next slide)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
21
Excel’s “z-Test: Two Sample for Means” Tool

Excel Dialog Box
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
22
Excel’s “z-Test: Two Sample for Means” Tool

1
2
3
4
5
6
7
8
9
10
11
12
13
Excel Value Worksheet
A
Par
255
270
294
245
300
262
281
257
268
295
249
291
B C
D
Rap
266
z-Test: Two Sample for Means
238
243
277
Mean
275
Known Variance
244
Observations
239
Hypothesized Mean Difference
242
z
280
P(Z<=z) one-tail
261
z Critical one-tail
276
P(Z<=z) two-tail
241
z Critical two-tail
E
Note: Rows
F are
14-121
not shown.
Par, Inc.
Rap, Ltd.
235
218
225
400
120
80
0
6.483545607
4.50145E-11
2.326341928
9.00291E-11
2.575834515
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
23
Inferences About the Difference Between
Two Population Means: s 1 and s 2 Unknown


Interval Estimation of m1 – m2
Hypothesis Tests About m1 – m2
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
24
Interval Estimation of m1 - m2:
s 1 and s 2 Unknown
When s 1 and s 2 are unknown, we will:
• use the sample standard deviations s1 and s2
as estimates of s 1 and s 2 , and
• replace z/2 with t/2.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
25
Interval Estimation of m1 - m2:
s 1 and s 2 Unknown

Interval Estimate
𝑥1 − 𝑥2 ± 𝑡𝛼/2
𝑠1 2 𝑠2 2
+
𝑛1
𝑛2
Where the degrees of freedom for ta/2 are:
2
𝑑𝑓 =
2 2
𝑠1
𝑠
+ 2
𝑛1
𝑛2
1
𝑠1 2
𝑛1 − 1 𝑛1
2
1
𝑠2 2
+
𝑛2 − 1 𝑛2
2
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
26
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Specific Motors of Detroit has developed a new
automobile known as the M car. 24 M cars and 28
J cars (from Japan) were road tested to compare
miles-per-gallon (mpg) performance. The sample
statistics are shown on the next slide.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
27
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Sample #1
M Cars
Sample Size
24 cars
Sample Mean
29.8 mpg
Sample Std. Dev. 2.56 mpg
Sample #2
J Cars
28 cars
27.3 mpg
1.81 mpg
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
28
Difference Between Two Population Means:
s 1 and s 2 Unknown

Example: Specific Motors
Let us develop a 90% confidence interval estimate
of the difference between the mpg performances of
the two models of automobile.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
29
Point Estimate of m 1 - m 2
Point estimate of m1 - m2 = 𝑥1 − 𝑥2
= 29.8 - 27.3
= 2.5 mpg
where:
m1 = mean miles-per-gallon for the
population of M cars
m2 = mean miles-per-gallon for the
population of J cars
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
30
Interval Estimation of m 1 - m 2:
s 1 and s 2 Unknown
The degrees of freedom for t/2 are:
𝑑𝑓 =
2
(2.56)2 (1.81)2
+ 28
24
2
2
1
(2.56)2
1
(1.81)2
+
24−1
24
28−1
28
= 40.59 = 41
With /2 = .05 and df = 41, t/2 = 1.683
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
31
Interval Estimation of m 1 - m 2:
s 1 and s 2 Unknown
𝑥1 − 𝑥2 ± 𝑡𝛼/2
𝑠1 2 𝑠2 2
+
𝑛1
𝑛2
(2.56)2 (1.81)2
29.8 − 27.3 ± 1.683
+
24
28
2.5 + 1.051 or
1.449 to 3.551 mpg
We are 90% confident that the difference between
the miles-per-gallon performances of M cars and J cars
is 1.449 to 3.551 mpg.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
32
Interval Estimation of m 1 - m 2:
s 1 and s 2 Unknown

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Excel Formula Worksheet
A
M
26.1
32.5
31.8
27.6
28.5
33.6
31.7
25.2
26.0
32.0
31.7
30.4
27.6
32.3
30.6
29.5
B
C
D
E
J
Par, Inc.
Rap, Ltd.
25.6
Sample Size =COUNT(A2:A25)
=COUNT(B2:B29)
28.1
Sample Mean =AVERAGE(A2:A25)
=AVERAGE(B2:B29)
27.9 Sample Std. Dev. =STDEV(A2:A25)
=STDEV(B2:B29)
25.3
30.1 Est. of Variance =D4^2/D2+E4^2/E2
27.5
Standard Error =SQRT(D6)
26.0
28.8
Confid. Coeff. 0.90
30.6
Level of Signif. =1-D9
24.4 Degr. of Freedom =D6^2/((1/(D2-1))*(D4^2/D2)^2+(1/(E2-1))*(E4^2/E2)^2))
27.3
t Value =T.INV.2T(D10,D11)
27.5
Margin of Error =D12*D7
26.3
Note:
25.5 Point Est. of Diff.=D3-E3
Rows 18-29
26.3
Lower Limit =D15-D13
are not shown.
24.3
Upper Limit =D15+D13
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
33
Interval Estimation of m 1 - m 2:
s 1 and s 2 Unknown

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
Excel Formula Worksheet
A
M
26.1
32.5
31.8
27.6
28.5
33.6
31.7
25.2
26.0
32.0
31.7
30.4
27.6
32.3
30.6
29.5
B
C
D
J
Par, Inc.
25.6
Sample Size 24
28.1
Sample Mean 29.8
27.9 Sample Std. Dev. 2.56
25.3
30.1 Est. of Variance 0.39007
27.5
Standard Error 0.62456
26.0
28.8
Confid. Coeff. 0.90
30.6
Level of Signif. 0.10
24.4 Degr. of Freedom 40.59.
27.3
t Value 1.683
27.5
Margin of Error 1.051
26.3
25.5 Point Est. of Diff.2.5
26.3
Lower Limit 1.449
24.3
Upper Limit 3.551
E
Rap, Ltd.
28
27.3
1.81
Note:
Note:
Rows
Rows 18-29
18-29
are
are not
not shown.
shown.
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
34
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown

Hypotheses
H0: m1 – m2 > D0 H0: m1 – m2 < D0 H0: m1 – m2 = D0
Ha: m1 – m2 < D0 Ha: m1 – m2 > D0 Ha: m1 – m2 ≠ D0
Left-tailed

Right-tailed
Two-tailed
Test Statistic
𝑡=
𝑥1 − 𝑥2 − 𝐷0
𝑠1 2 𝑠2 2
+
𝑛1
𝑛2
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
35
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown

Example: Specific Motors
Can we conclude, using a .05 level of significance,
that the miles-per-gallon (mpg) performance of M cars
is greater than the miles-per-gallon performance of J
cars?
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
36
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown
 p–Value and Critical Value Approaches
1. Develop the hypotheses.
H0: m1 - m2 < 0
Ha: m1 - m2 > 0
Righttailed
test
where:
m1 = mean mpg for the population of M cars
m2 = mean mpg for the population of J cars
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
37
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown
 p–Value and Critical Value Approaches
2. Specify the level of significance.  = .05
3. Compute the value of the test statistic.
𝑡=
𝑥1 − 𝑥2 − 𝐷0
2
2
𝑠1
𝑠2
+
𝑛1
𝑛2
=
29.8 − 27.3 − 0
(2.56)2 (1.81)2
+
24
28
= 4.003
© 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted
in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
38
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown
 p–Value Approach
4. Compute the p –value.
The degrees of freedom for t are:
𝑑𝑓 =
2
(2.56)2 (1.81)2
+ 28
24
2
2
1
(2.56)2
1
(1.81)2
+28−1 28
24−1
24
= 40.59 = 41
With /2 = .05 and df = 41, t/2 = 1.683
Because t = 4.003 > t.005 = 1.683, the p–value < .005.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
39
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown
 p-Value Approach
5. Determine whether to reject H0.
Because p-value <  = .05, we reject H0.
We are at least 95% confident that the miles-pergallon (mpg) performance of M cars is greater than
the miles-per-gallon performance of J cars.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
40
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown
 Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .05 and df = 41, t.05 = 1.683
Reject H0 if t > 1.683
5. Determine whether to reject H0.
Because 4.003 > 1.683, we reject H0.
We are at least 95% confident that the miles-pergallon (mpg) performance of M cars is greater than
the miles-per-gallon performance of J cars.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
41
Excel’s “z-Test: Two-Sample
Assuming Unequal Variances” Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose t-Test: Two-Sample Assuming
Unequal Variances from the list of
Analysis Tools
Step 4 When the t-Test: Two-Sample Assuming
Unequal Variances dialog box appears:
(see details on next slide)
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
42
Hypothesis Tests About m 1 - m 2:
s 1 and s 2 Unknown

Excel Dialog Box
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
43
Excel’s “z-Test: Two-Sample
Assuming Unequal Variances” Tool

Excel Value Worksheet
A
1
2
3
4
5
6
7
8
9
10
11
12
13
B
Mcar Jcar
26.1
32.5
31.8
27.6
28.5
33.6
31.7
25.2
26.0
32.0
31.7
30.4
25.6
28.1
27.9
25.3
30.1
27.5
26.0
28.8
30.6
24.4
27.3
27.5
C
D
E
F
z-Test: Two-Sample Assuming Unequal Variances
Mean
Variance
Observations
Hypothesized Mean Difference
df
t Stat
P(T<=t) one-tail
t Critical one-tail
P(T<=t) two-tail
t Critical two-tail
Mcar
29.79583
6.555199
24
0
41
3.99082
1.33000E-04
1.682879
0.000266
2.019542
Jcar
27.30357
3.272209
28
Note:
Rows
14-121
are not
shown.
Note: Rows 14-121 are not shown.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
44
Inferences About the Difference Between
Two Population Means: Matched Samples
 With a matched-sample design each sampled item
provides a pair of data values.
 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.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
45
Inferences About the Difference Between
Two Population Means: Matched Samples

Example: Express Deliveries
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.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
46
Inferences About the Difference Between
Two Population Means: Matched Samples

Example: Express Deliveries
In testing the delivery times of the two services,
the firm sent two reports to a random sample of its
district offices with one report carried by UPX and
the other report carried by INTEX. Do the data on
the next slide indicate a difference in mean delivery
times for the two services? Use a .05 level of
significance.
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
47
Inferences About the Difference Between
Two Population Means: Matched Samples
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
25
24
15
15
13
15
15
8
9
11
7
6
4
1
2
3
-1
2
-2
5
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
48
Inferences About the Difference Between
Two Population Means: Matched Samples
 p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: md = 0
Ha: md  0
Let md = the mean of the difference values for the
two delivery services for the population
of district offices
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
49
Inferences About the Difference Between
Two Population Means: Matched Samples
 p –Value and Critical Value Approaches
2. Specify the level of significance.  = .05
3. Compute the value of the test statistic.
𝑑𝑖
7 +6 +⋯+5
𝑑=
=
= 2.7
𝑛
10
𝑑𝑖 − 𝑑
𝑛−1
𝑠𝑑 =
𝑡=
𝑑−𝜇𝑑
𝑠𝑑
𝑛
2.7−0
=2.9
2
=
76.1
= 2.9
9
= 2.944
10
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
50
Inferences About the Difference Between
Two Population Means: Matched Samples
 p –Value Approach
4. Compute the p-value.
For t = 2.944 and df = 9, the p-value is between
.02 and .01. (This is a two-tailed test, so we
double the upper-tail areas of .01 and .005.)
5. Determine whether to reject H0.
Because p–value <  = .05, we reject H0.
We are at least 95% confident that there is a
difference in mean delivery times for the two
services?
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
51
Inferences About the Difference Between
Two Population Means: Matched Samples
 Critical Value Approach
4. Determine the critical value and rejection rule.
For  = .05 and df = 9, t.025 = 2.262.
Reject H0 if t > 2.262
5. Determine whether to reject H0.
Because t = 2.944 > 2.262, we reject H0.
We are at least 95% confident that there is a
difference in mean delivery times for the two
services?
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
52
Inferences About the Difference Between
Two Population Means: Matched Samples

Excel’s “t-Test: Paired Two Sample for Means” Tool
Step 1 Click the Data tab on the Ribbon
Step 2 In the Analysis group, click Data Analysis
Step 3 Choose t-Test: Paired Two Sample for Means
from the list of Analysis Tools
… continued
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
53
Inferences About the Difference Between
Two Population Means: Matched Samples

Excel Dialog Box
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
54
Inferences About the Difference Between
Two Population Means: Matched Samples

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Excel Value Worksheet
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
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
55
End of Chapter 10, Part A
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in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.
56