1. Business and Economics 9th Edition
Statistics for
Business and Economics 9th Edition
Chapter 10
Statistical Inferences About Means and Proportions with Two Populations

2. Two Sample Tests Two Sample Tests
Population Means, Independent Samples
Population Means, Matched Pairs
在第8章和第9章我们讲述了如何对一个总体均值及比例进行区间估计和假设检验。在本章中，我们将继续统计推断问题的讨论，讲述当两个总体均值或比例之差很重要的情况下，如何进行两个总体情形的区间估计和假设检验。
Population Proportions
Interval
Estimation
Examples:
Same group before vs. after treatment
Group 1 vs. independent Group 2
Hypothesis Tests
Proportion 1 vs. Proportion 2

3. Difference Between Two Means
Population means, independent samples
σx2 and σy2 known
Test statistic is a z value
σx2 and σy2 unknown
大样本
σx2 and σy2 assumed equal
Test statistic is a a value from the Student’s t distribution
小样本
σx2 and σy2 assumed unequal

4. Difference Between Two Means
Goal: Form a confidence interval for the difference between two population means, μx – μy
Population means, independent samples
独立简单随机样本（independent simple random samples）：总体1中的样本与总体2中的样本都是独立抽取的
Different data sources
Unrelated
Independent
Sample selected from one population has no effect on the sample selected from the other population

5. Population means, independent samples
σx2 and σy2 Known
Population means, independent samples
Assumptions:
Samples are randomly and
independently drawn
both population distributions
are normal
Population variances are
known *
σx2 and σy2 known
σx2 and σy2 unknown

6. Estimating the Difference Between Two Population Means
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 equal the mean of sample 1 and equal the
mean of sample 2.
The point estimator of the difference between the
means of the populations 1 and 2 is

7. Sampling Distribution of
Expected Value
Standard Deviation (Standard Error)
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

8. Interval Estimation of 1 - 2: s 1 and s 2 Known
Interval Estimate
where:
1 -  is the confidence coefficient
在两个总体方差已知的情况下，大样本和小样本（假设两总体都是正态分布），都可以通过标准正态分布来求 样本区间。

9. Difference Between Two Means
(continued)
Population means, independent samples
σx2 and σy2 known
Test statistic is a z value
σx2 and σy2 unknown
大样本
σx2 and σy2 assumed equal
Test statistic is a a value from the Student’s t distribution
小样本
σx2 and σy2 assumed unequal

10. Population means, independent samples
σx2 and σy2 Known
Population means, independent samples
Assumptions:
Samples are randomly and
independently drawn
both population distributions
are normal
Population variances are
known *
σx2 and σy2 known
σx2 and σy2 unknown

11. Population means, independent samples
σx2 and σy2 Known
(continued)
When σx2 and σy2 are known and both populations are normal, the variance of X – Y is
Population means, independent samples *
σx2 and σy2 known
…and the random variable
has a standard normal distribution
σx2 and σy2 unknown

12. Test Statistic, σx2 and σy2 Known
Population means, independent samples
The test statistic for
μx – μy is: *
σx2 and σy2 known
σx2 and σy2 unknown

13. Interval Estimation of 1 - 2: s 1 and s 2 Known
Interval Estimate
where:
1 -  is the confidence coefficient
在两个总体方差已知的情况下，大样本和小样本（假设两总体都是正态分布），都可以通过标准正态分布来求 样本区间。

14. Hypothesis Tests for Two Population Means
Two Population Means, Independent Samples
Lower-tail test:
H0: μx  μy
H1: μx < μy
i.e.,
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx ≤ μy
H1: μx > μy
i.e.,
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx = μy
H1: μx ≠ μy
i.e.,
H0: μx – μy = 0
H1: μx – μy ≠ 0

15. Two Population Means, Independent Samples, Variances Known
Decision Rules
Two Population Means, Independent Samples, Variances Known
Lower-tail test:
H0: μ1 – μ2  0
H1: μ1 – μ2 < 0
Upper-tail test:
H0: μ1 – μ2 ≤ 0
H1: μ1 – μ2 > 0
Two-tail test:
H0: μ1 – μ2= 0
H1: μ1 – μy ≠ 0 a a
a/2
a/2
-za za
-za/2
za/2
Reject H0 if z < -za
Reject H0 if z > za
Reject H0 if z < -za/2
or z > za/2

16. Interval Estimation of 1 - 2: 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.

17. Interval Estimation of 1 - 2: s 1 and s 2 Known
Example: Par, Inc.
Sample #1
Par, Inc.
Sample #2
Rap, Ltd.
Sample Size
120 balls balls
Sample Mean
275 yards 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.

18. Interval Estimation of 1 - 2: 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.

19. Estimating the Difference Between Two Population Means
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
x1 = sample mean distance
for the Par golf balls
Simple random sample
of n2 Rap golf balls
x2 = sample mean distance
for the Rap golf balls
x1 - x2 = Point Estimate of m1 – m2

20. Point Estimate of 1 - 2 Point estimate of 1 - 2 = = 275 - 258
= 17 yards
where:
1 = mean distance for the population
of Par, Inc. golf balls
2 = mean distance for the population
of Rap, Ltd. golf balls

21. Interval Estimation of 1 - 2:  1 and  2 Known
or yards to yards
We are 95% confident that the difference between
the mean driving distances of Par, Inc. balls and Rap,
Ltd. balls is to yards.

22. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Known
Example: Par, Inc.
Can we conclude, using
a = .01, that the mean driving
distance of Par, Inc. golf balls
is greater than the mean driving
distance of Rap, Ltd. golf balls?

23. Hypothesis Tests About m 1 - m 2: s 1 and s 2 Known
p –Value and Critical Value Approaches
1. Develop the hypotheses.
H0: 1 - 2 < 0 
Ha: 1 - 2 > 0
where:
1 = mean distance for the population
of Par, Inc. golf balls
2 = mean distance for the population
of Rap, Ltd. golf balls
2. Specify the level of significance.
a = .01

24. 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.

25. 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 <
5. Determine whether to reject H0.
Because p–value < a = .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.

26. 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 a = .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.

27. Population means, independent samples
σx2 and σy2 Unknown,
Population means, independent samples
σx2 and σy2 known
σx2 and σy2 unknown *
大样本
小样本

28. Test Statistic, σx2 and σy2 Unknown, Equal
大样本
（nx>=30且ny>=30） *
合并方差估计量是两个样本方差的加权平均，其取值点介于s1和是之间
小样本（nx<30或ny<30）
σx2 and σy2 assumed equal
σx2 and σy2 assumed unequal

29. Test Statistic, σx2 and σy2 Unknown, Equal
且 小样本
σx2 and σy2 unknown
Assumptions:
Samples are randomly and
independently drawn
Populations are normally
distributed
Population variances are
unknown but assumed equal *
σx2 and σy2 assumed equal
合并方差估计量是两个样本方差的加权平均，其取值点介于s1和是之间
σx2 and σy2 assumed unequal

30. Test Statistic, σx2 and σy2 Unknown, Equal
且 小样本
σx2 and σy2 unknown
Forming interval estimates:
The population variances
are assumed equal, so use
the two sample standard
deviations and pool them to
estimate σ
use a t value with
(nx + ny – 2) degrees of
freedom *
σx2 and σy2 assumed equal
合并方差估计量是两个样本方差的加权平均，其取值点介于s1和是之间
σx2 and σy2 assumed unequal

31. Test Statistic, σx2 and σy2 Unknown, Equal
且 小样本
σx2 and σy2 unknown
σ 2的合并估计量： *
σx2 and σy2 assumed equal
合并方差估计量是两个样本方差的加权平均，其取值点介于s1和是之间
σx2 and σy2 assumed unequal

32. Test Statistic, σx2 and σy2 Unknown, Equal
且 小样本
σx2 and σy2 unknown
The test statistic for
μx – μy is: *
σx2 and σy2 assumed equal
T分布并不局限于小样本情形，只要两个分布都服从服从正态分布且总体方差相等，它都是可行的。只是在大样本情形下，我们不需要使用t分布，用标准正态分布就可以了。
σx2 and σy2 assumed unequal
Where t has (n1 + n2 – 2) d.f.,
and

33. Interval Estimation, σx2 and σy2 Unknown, Equal
且 小样本
σx2 and σy2 unknown *
σx2 and σy2 assumed equal
T分布并不局限于小样本情形，只要两个分布都服从服从正态分布且总体方差相等，它都是可行的。只是在大样本情形下，我们不需要使用t分布，用标准正态分布就可以了。
σx2 and σy2 assumed unequal
Where t has (n1 + n2 – 2) d.f.,
and

34. σx2 and σy2 Unknown, Assumed Unequal
且 小样本
σx2 and σy2 unknown
Assumptions:
Samples are randomly and
independently drawn
Populations are normally
distributed
Population variances are
unknown and assumed
unequal
σx2 and σy2 assumed equal
T分布并不局限于小样本情形，只要两个分布都服从服从正态分布且总体方差相等，它都是可行的。只是在大样本情形下，我们不需要使用t分布，用标准正态分布就可以了。 *
σx2 and σy2 assumed unequal

35. σx2 and σy2 Unknown, Assumed Unequal
且 小样本
σx2 and σy2 unknown
Forming interval estimates:
The population variances are
assumed unequal, so a pooled
variance is not appropriate
use a t value with  degrees
of freedom, where
σx2 and σy2 assumed equal
T分布并不局限于小样本情形，只要两个分布都服从服从正态分布且总体方差相等，它都是可行的。只是在大样本情形下，我们不需要使用t分布，用标准正态分布就可以了。 *
σx2 and σy2 assumed unequal

36. Test Statistic, σx2 and σy2 Unknown, Unequal
且 小样本
σx2 and σy2 unknown
The test statistic for
μx – μy is:
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal
Where t has  degrees of freedom:

37. Interval Estimation, σx2 and σy2 Unknown, Unequal
且 小样本
σx2 and σy2 unknown
σx2 and σy2 assumed equal *
σx2 and σy2 assumed unequal
Where t has  degrees of freedom:

38. Two Population Means, Independent Samples, Variances Unknown
Decision Rules
Two Population Means, Independent Samples, Variances Unknown
Lower-tail test:
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0 a a
a/2
a/2
-ta ta
-ta/2
ta/2
Reject H0 if t < -tn-1, a
Reject H0 if t > tn-1, a
Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
Where t has n - 1 d.f.

39. Pooled Variance t Test: Example
You are a financial analyst for a brokerage firm. Is there a difference in dividend yield between stocks listed on the NYSE & NASDAQ? You collect the following data:
NYSE NASDAQ Number
Sample mean
Sample std dev
纽约股票交易所（NYSE）纳斯达克(NASDAQ)是指美国"全国证券交易商协会自动报价系统"，
Assuming both populations are
approximately normal with equal variances, is there a difference in average yield ( = 0.05)?

40. Calculating the Test Statistic
The test statistic is:

41. Solution t Decision: Reject H0 at a = 0.05 Conclusion:
H0: μ1 - μ2 = 0 i.e. (μ1 = μ2)
H1: μ1 - μ2 ≠ 0 i.e. (μ1 ≠ μ2)
 = 0.05
df = = 44
Critical Values: t = ±
Test Statistic:
.025
.025
2.0154 t
2.040
Decision:
Conclusion:
Reject H0 at a = 0.05
There is evidence of a difference in means.

42. Two Sample Tests Two Sample Tests
Population Means, Independent Samples
Population Means, Matched Pairs
在第8章和第9章我们讲述了如何对一个总体均值及比例进行区间估计和假设检验。在本章中，我们将继续统计推断问题的讨论，讲述当两个总体均值或比例之差很重要的情况下，如何进行两个总体情形的区间估计和假设检验。
Population Proportions
Interval
Estimation
Examples:
Same group before vs. after treatment
Group 1 vs. independent Group 2
Hypothesis Tests
Proportion 1 vs. Proportion 2

43. Matched Pairs Tests Means of 2 Related Populations di = xi - yi
Paired or matched samples
Repeated measures (before/after)
Use difference between paired values:
Assumptions:
Both Populations Are Normally Distributed
Matched Pairs
匹配样本方案：在匹配样本方案中，两种生产方法是在相同条件下接受检验的，的以该方案会比独立样本方案产生更小的误差。
di = xi - yi

44. Test Statistic: Matched Pairs
The test statistic for the mean difference is a t value, with
n – 1 degrees of freedom:
Matched Pairs
Where
D0 = hypothesized mean difference
sd = sample standard dev. of differences
n = the sample size (number of pairs)

45. Decision Rules: Matched Pairs
Paired Samples
Lower-tail test:
H0: μx – μy  0
H1: μx – μy < 0
Upper-tail test:
H0: μx – μy ≤ 0
H1: μx – μy > 0
Two-tail test:
H0: μx – μy = 0
H1: μx – μy ≠ 0 a a
a/2
a/2
-ta ta
-ta/2
ta/2
Reject H0 if t < -tn-1, a
Reject H0 if t > tn-1, a
Reject H0 if t < -tn-1 , a/2
or t > tn-1 , a/2
has n - 1 d.f.
Where

46. Matched Pairs Example  d = = - 4.2
Assume you send your salespeople to a “customer service” training workshop. Has the training made a difference in the number of complaints? You collect the following data:  di
d =
Number of Complaints: (2) - (1)
Salesperson Before (1) After (2) Difference, di
C.B
T.F
M.H
R.K
M.O
-21 n
= - 4.2

47. Critical Value = ± 4.604 d.f. = n - 1 = 4
Matched Pairs: Solution
Has the training made a difference in the number of complaints (at the  = 0.01 level)?
Reject
Reject
H0: μx – μy = 0
H1: μx – μy  0
/2
/2
 = .01
d =
- 4.2
- 1.66
Critical Value = ± d.f. = n - 1 = 4
Decision: Do not reject H0
(t stat is not in the reject region)
Test Statistic:
Conclusion: There is not a significant change in the number of complaints.

48. Two Sample Tests Two Sample Tests
Population Means, Independent Samples
Population Means, Matched Pairs
在第8章和第9章我们讲述了如何对一个总体均值及比例进行区间估计和假设检验。在本章中，我们将继续统计推断问题的讨论，讲述当两个总体均值或比例之差很重要的情况下，如何进行两个总体情形的区间估计和假设检验。
Population Proportions
Interval
Estimation
Examples:
Same group before vs. after treatment
Group 1 vs. independent Group 2
Hypothesis Tests
Proportion 1 vs. Proportion 2

49. Two Population Proportions
Goal: Test hypotheses for the difference between two population proportions, P1 – P2
Population proportions
Assumptions:
Both sample sizes are large,

50. Two Population Proportions

51. Interval Estimation for Two Population Proportions

52. Test Statistic for Two Population Proportions
The test statistic for
H0: P1 – P2 = 0
is a z value:
Population proportions
Where

53. Decision Rules: Proportions
Population proportions
Lower-tail test:
H0: p1 – p2  0
H1: p1 – p2 < 0
Upper-tail test:
H0: p1 – p2 ≤ 0
H1: p1 – p2 > 0
Two-tail test:
H0: p1– p2 = 0
H1: p1 – p2 ≠ 0 a a
a/2
a/2
-za za
-za/2
za/2
Reject H0 if z < -za
Reject H0 if z > za
Reject H0 if z < -za/2
or z > za/2

54. Example: Two Population Proportions
Is there a significant difference between the proportion of men and the proportion of women who will vote Yes on Proposition A?
In a random sample, 36 of 72 men and 31 of 50 women indicated they would vote Yes
Test at the .05 level of significance

55. Example: Two Population Proportions
(continued)
The hypothesis test is:
H0: PM – PW = 0 (the two proportions are equal)
H1: PM – PW ≠ 0 (there is a significant difference between
proportions)
The sample proportions are:
Men: = 36/72 = .50
Women: = 31/50 = .62
The estimate for the common overall proportion is:

56. Example: Two Population Proportions
(continued)
Reject H0
Reject H0
The test statistic for PM – PW = 0 is:
.025
.025
-1.96
1.96
-1.31
Decision: Do not reject H0
Conclusion: There is not significant evidence of a difference between men and women in proportions who will vote yes.
Critical Values = ±1.96
For  = .05

57. Chapter Summary Compared two dependent samples (paired samples)
Performed paired sample t test for the mean difference
Compared two independent samples
Performed z test for the differences in two means
Performed pooled variance t test for the differences in two means
Compared two population proportions
Performed z-test for two population proportions