1. STATISTICS INFORMED DECISIONS USING DATA
Fifth Edition
Chapter 11
Inferences on Two Samples
Copyright © 2018, 2014, 2011 Pearson Education, Inc. All Rights Reserved

2. 11.1 Inference about Two Population Proportions Learning Objectives
1. Distinguish between independent and dependent sampling 2. Test hypotheses regarding two proportions from independent samples 3. Construct and interpret confidence intervals for the difference between two population proportions 4. Determine the sample size necessary for estimating the difference between two population proportions

3. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Distinguish between Independent and Dependent Sampling (1 of 4)
A sampling method is independent when the individuals selected for one sample do not dictate which individuals are to be in a second sample. A sampling method is dependent when the individuals selected to be in one sample are used to determine the individuals to be in the second sample.

4. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Distinguish between Independent and Dependent Sampling (2 of 4)
Dependent samples are often referred to as matched-pairs samples. It is possible for an individual to be matched against him- or herself.

5. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Distinguish between Independent and Dependent Sampling (3 of 4)
Parallel Example 1: Distinguish between Independent and Dependent Sampling
For each of the following, determine whether the sampling method is independent or dependent.
A researcher wants to know whether the price of a one night stay at a Holiday Inn Express is less than the price of a one night stay at a Red Roof Inn. She randomly selects 8 towns where the location of the hotels is close to each other and determines the price of a one night stay.
A researcher wants to know whether the “state” quarters (introduced in 1999) have a mean weight that is different from “traditional” quarters. He randomly selects 18 “state” quarters and 16 “traditional” quarters and compares their weights.

6. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Distinguish between Independent and Dependent Sampling (4 of 4)
Solution
The sampling method is dependent since the 8 Holiday Inn Express hotels can be matched with one of the 8 Red Roof Inn hotels by town.
The sampling method is independent since the “state” quarters which were sampled had no bearing on which “traditional” quarters were sampled.

7. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (1 of 30)
Sampling Distribution of the Difference between Two Proportions (Independent Sample)

8. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (2 of 30)
Sampling Distribution of the Difference between Two Proportions (Independent Sample)

9. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (3 of 30)
Sampling Distribution of the Difference between Two Proportions

10. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (4 of 30)

11. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (5 of 30)
Hypothesis Test Regarding the Difference between Two Population Proportions
To test hypotheses regarding two population proportions, p1 and p2, we can use the steps that follow, provided that:
the samples are independently obtained using simple random sampling,

12. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (6 of 30)
Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways:
Two-Tailed
Left-Tailed
Right-Tailed
H0: p1 = p2
H1: p1 ≠ p2
H1: p1 < p2
H1: p1 > p2
Note: p1 is the population proportion for population 1, and p2 is the population proportion for population 2.

13. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (7 of 30)
Step 2: Select a level of significance, α, depending on the seriousness of making a Type I error.

14. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (8 of 30)
Classical Approach
Step 3: Compute the test statistic

15. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (9 of 30)
Classical Approach
Use Table V to determine the critical value.

16. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (10 of 30)
Classical Approach
Two-Tailed

17. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (11 of 30)
Classical Approach
Left-Tailed

18. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (12 of 30)
Classical Approach
Right-Tailed

19. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (13 of 30)
P-value Approach
By Hand Step 3: Compute the test statistic

20. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (14 of 30)
P-value Approach
Use Table V to determine the P-Value.

21. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (15 of 30)
P-value Approach
Two-Tailed

22. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (16 of 30)
P-value Approach
Left-Tailed

23. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (17 of 30)
P-value Approach
Right-Tailed

24. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (18 of 30)
P-value Approach
Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, Excel, MINITAB, and StatCrunch are in the Technology Step-by- Step in the text.

25. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (19 of 30)
Classical Approach
Step 4: Compare the critical value with the test statistic:

26. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (20 of 30)
P-value Approach
Step 4: If P-value < α, reject the null hypothesis.

27. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (21 of 30)
P-value Approach
Step 5: State the conclusion.

28. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (22 of 30)
Parallel Example 1: Testing Hypotheses Regarding Two Population Proportions
An economist believes that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. He obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Test the economist’s claim at the α = 0.05 level of significance.

29. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (23 of 30)
Solution
We must first verify that the requirements are satisfied:
1. The samples are simple random samples that were obtained independently. 2.

30. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (24 of 30)
Solution
Step 1: We want to determine whether the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access. So,
H0: p1 = p2 versus H1: p1 > p2
or, equivalently,
H0: p1 − p2 = 0 versus H1: p1 − p2 > 0
Step 2: The level of significance is α = 0.05.

31. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (25 of 30)
Solution

32. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (26 of 30)
Solution: Classical Approach
This is a right-tailed test with α = The critical value is z0.05 =

33. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (27 of 30)
Step 4: Since the test statistic, z0 = 1.33 is less than the critical value z.05 = 1.645, we fail to reject the null hypothesis.

34. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (28 of 30)
Solution: P-value Approach
Because this is a right-tailed test, the P-value is the area under the normal to the right of the test statistic z0 = That is, P-value = P(Z > 1.33) ≈ 0.09.

35. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (29 of 30)
Solution: P-value Approach
Step 4: Since the P-value is greater than the level of significance α = 0.05, we fail to reject the null hypothesis.

36. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Test Hypotheses Regarding Two Population Proportions from Independent Samples (30 of 30)
Solution
Step 5: There is insufficient evidence at the α = 0.05 level to conclude that the percentage of urban households with Internet access is greater than the percentage of rural households with Internet access.

37. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions (1 of 6)
Constructing a (1 − α) • 100% Confidence Interval for the Difference between Two Population Proportions
To construct a (1 − α) • 100% confidence interval for the difference between two population proportions, the following requirements must be satisfied:
the samples are obtained independently using simple random sampling,

38. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions (2 of 6)
Constructing a (1 − α) • 100% Confidence Interval for the Difference between Two Population Proportions
Provided that these requirements are met, a (1 − α) • 100% confidence interval for p1 − p2 is given by

39. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions (3 of 6)
Parallel Example 3: Constructing a Confidence Interval for the Difference between Two Population Proportions
An economist obtains a random sample of 800 urban households and finds that 338 of them have Internet access. He obtains a random sample of 750 rural households and finds that 292 of them have Internet access. Find a 99% confidence interval for the difference between the proportion of urban households that have Internet access and the proportion of rural households that have Internet access.

40. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions (4 of 6)
Solution
We have already verified the requirements for constructing a confidence interval for the difference between two population proportions in the previous example.

41. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions (5 of 6)
Solution
Thus,

42. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Construct and Interpret Confidence Intervals for the Difference between Two Population Proportions (6 of 6)
Solution
We are 99% confident that the difference between the proportion of urban households that have Internet access and the proportion of rural households that have Internet access is between −0.03 and Since the confidence interval contains 0, we are unable to conclude that the proportion of urban households with Internet access is greater than the proportion of rural households with Internet access.

43. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Determine the Sample Size Necessary for Estimating the Difference between Two Population Proportions (1 of 4)
Sample Size for Estimating p1 − p2
The sample size required to obtain a (1 − α)•100% confidence interval with a margin of error, E, is given by

44. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Determine the Sample Size Necessary for Estimating the Difference between Two Population Proportions (2 of 4)
Parallel Example 5: Determining Sample Size
A doctor wants to estimate the difference in the proportion of year old mothers that received prenatal care and the proportion of year old mothers that received prenatal care. What sample size should be obtained if she wished the estimate to be within 2 percentage points with 95% confidence assuming:
she uses the results of the National Vital Statistics Report results in which 98% of the year old mothers received prenatal care and 99.2% of year old mothers received prenatal care.
she does not use any prior estimates.

45. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Determine the Sample Size Necessary for Estimating the Difference between Two Population Proportions (3 of 4)
The doctor must sample 265 randomly selected year old mothers and 265 randomly selected year old mothers.

46. 11. 1 Inference about Two Population Proportions 11. 1
11.1 Inference about Two Population Proportions Determine the Sample Size Necessary for Estimating the Difference between Two Population Proportions (4 of 4)
The doctor must sample 4802 randomly selected year old mothers and 4802 randomly selected year old mothers. Note that having prior estimates of p1 and p2 reduces the number of mothers that need to be surveyed.

47. 11.2 Inference about Two Means: Dependent Samples Learning Objectives
1. Test hypotheses for a population mean from matched-pairs data 2. Construct and interpret confidence intervals about the population mean difference of matched-pairs data

48. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (1 of 30)
“In Other Words”
Statistical inference methods on matched-pairs data use the same methods as inference on a single population mean, except that the differences are analyzed.

49. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (2 of 30)
Testing Hypotheses Regarding the Difference of Two Means Using a Matched-Pairs Design
To test hypotheses regarding the mean difference of data obtained form a dependent sample (matched-pairs data), use the following steps. provided that
the sample is obtained using simple random sampling,
the sample data are matched pairs,
the differences are normally distributed with no outliers or the sample size, n, is large (n ≥ 30),
the sampled values are independent

50. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (3 of 30)
Step 1: Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways, where μd is the population mean difference of the matched-pairs data.
Two-Tailed
Left-Tailed
Right-Tailed
H0: µd = 0
H1: µd ≠ 0
H1: µd < 0
H1: µd > 0

51. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (4 of 30)
Step 2: Select a level of significance, α, depending on the seriousness of making a Type I error.

52. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (5 of 30)
Classical Approach
Step 3: Compute the test statistic

53. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (6 of 30)
Classical Approach
Use Table VII to determine the critical value using n − 1 degrees of freedom.

54. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (7 of 30)
Classical Approach
Two-Tailed

55. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (8 of 30)
Classical Approach
Left-Tailed

56. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (9 of 30)
Classical Approach
Right-Tailed

57. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (10 of 30)
Classical Approach
Step 4: Compare the critical value with the test statistic:

58. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (11 of 30)
P-value Approach
Step 3: Compute the test statistic

59. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (12 of 30)
P-value Approach
Use Table VII to approximate the P-value using n − 1 degrees of freedom.

60. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (13 of 30)
P-value Approach
Two-Tailed

61. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (14 of 30)
P-value Approach
Left-Tailed

62. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (15 of 30)
P-value Approach
Right-Tailed

63. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (16 of 30)
P-value Approach
Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, MINITAB, Excel and StatCrunch, are in the Technology Step-by- Step in the text.

64. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (17 of 30)
P-value Approach
Step 4: If P-value < α, reject the null hypothesis.

65. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (18 of 30)
P-value Approach
Step 5: State the conclusion.

66. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (19 of 30)
These procedures are robust, which means that minor departures from normality will not adversely affect the results. However, if the data have outliers, the procedure should not be used.

67. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (20 of 30)
Parallel Example 2: Testing a Claim Regarding Matched- Pairs Data
The following data represent the cost of a one night stay in Hampton Inn Hotels and La Quinta Inn Hotels for a random sample of 10 cities. Test the claim that Hampton Inn Hotels are priced differently than La Quinta Hotels at the α = 0.05 level of significance.

68. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (21 of 30)
City
Hampton Inn
La Quinta
Dallas
129
105
Tampa Bay
149 96
St. Louis 49
Seattle
189
San Diego
109
119
Chicago
160 89
New Orleans 72
Phoenix 59
Atlanta 90
Orlando 69

69. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (22 of 30)
Solution
This is a matched-pairs design since the hotel prices come from the same ten cities. To test the hypothesis, we first compute the differences and then verify that the differences come from a population that is approximately normally distributed with no outliers because the sample size is small.

70. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (23 of 30)
Solution

71. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (24 of 30)
Solution

72. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (25 of 30)
Solution
Step 1: We want to determine if the prices differ:
H0: μd = 0 versus H1: μd ≠ 0
Step 2: The level of significance is α = 0.05.
Step 3: The test statistic is

73. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (26 of 30)
Solution: Classical Approach
This is a two-tailed test so the critical values at the α = 0.05 level of significance with n − 1 = 10 − 1 = 9 degrees of freedom are −t0.025 = −2.262 and t0.025 =

74. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (27 of 30)
Solution: Classical Approach
Step 4: Since the test statistic, t0 = 5.27 is greater than the critical value t.025 = 2.262, we reject the null hypothesis.

75. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (28 of 30)
Solution: P-value Approach
Because this is a two-tailed test, the P-value is two times the area under the t-distribution with n − 1 = 10 − 1 = 9 degrees of freedom to the right of the test statistic t0 = 5.27.
That is, P-value = 2P(t > 5.27) ≈ 2( ) = (using technology). Approximately 5 samples in 10,000 will yield results at least as extreme as we obtained if the null hypothesis is true.

76. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (29 of 30)
Solution: P-value Approach
Step 4: Since the P-value is less than the level of significance α = 0.05, we reject the null hypothesis.

77. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Test Hypotheses for a Population Mean from Matched-Pairs Data (30 of 30)
Solution
Step 5: There is sufficient evidence to conclude that Hampton Inn hotels and La Quinta hotels are priced differently at the α = 0.05 level of significance.

78. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data (1 of 5)
Confidence Interval for Matched-Pairs Data
A (1 − α) • 100% confidence interval for μd is given by

79. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data (2 of 5)
Confidence Interval for Matched-Pairs Data
Note: The interval is exact when the population is normally distributed and approximately correct for nonnormal populations, provided that n is large.

80. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data (3 of 5)
Parallel Example 4: Constructing a Confidence Interval for Matched-Pairs Data
Construct a 90% confidence interval for the mean difference in price of Hampton Inn versus La Quinta hotel rooms.

81. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data (4 of 5)
Solution
We have already verified that the differenced data come from a population that is approximately normal with no outliers.

82. 11. 2 Inference about Two Means: Dependent Samples 11. 2
11.2 Inference about Two Means: Dependent Samples Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data (5 of 5)
We are 90% confident that the mean difference in hotel room price for Hampton Inn versus La Quinta Inn is between $33.53 and $69.27.

83. 11.3 Inference about Two Means: Independent Samples Learning Objectives
1. Test hypotheses regarding the difference of two independent means 2. Construct and interpret confidence intervals regarding the difference of two independent means

84. 11.3 Inference about Two Means: Independent Samples Introduction (1 of 2)
Sampling Distribution of the Difference of Two Means: Independent Samples with Population
Standard Deviations Unknown (Welch’s t)
Suppose that a simple random sample of size n1 is taken from a population with unknown mean μ1 and unknown standard deviation σ1. In addition, a simple random sample of size n2 is taken from a population with unknown mean μ2 and unknown standard deviation σ2. If the two populations are normally distributed or the sample sizes are sufficiently large (n1 ≥ 30, n2 ≥ 30), then

85. 11.3 Inference about Two Means: Independent Samples Introduction (2 of 2)
Sampling Distribution of the Difference of Two Means: Independent Samples with Population
Standard Deviations Unknown (Welch’s t)

86. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (1 of 29)
Testing Hypotheses Regarding the Difference of Two Means
To test hypotheses regarding two population means, μ1 and μ2, with unknown population standard deviations, we can use the following steps, provided that:
the samples are obtained using simple random sampling;
the samples are independent;
the populations from which the samples are drawn are normally distributed or the sample sizes are large (n1 ≥ 30, n2 ≥ 30);
for each sample, the sample size is no more than 5% of the population size.

87. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (2 of 29)
Step 1: Determine the null and alternative hypotheses. The hypotheses are structured in one of three ways:
Two-Tailed
Left-Tailed
Right-Tailed
H0: µ1 = µ2
H1: µ1 ≠ µ2
H1: µ1 < µ2
H1: µ1 > µ2
Note: µ1 is the population mean for population 1, and µ2 is the population mean for population 2.

88. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (3 of 29)
Step 2: Select a level of significance, α, depending on the seriousness of making a Type I error.

89. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (4 of 29)
Classical Approach
Step 3: Compute the test statistic

90. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (5 of 29)
Classical Approach
Use Table VII to determine the critical value using the smaller of n1 − 1 or n2 − 1 degrees of freedom.

91. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (6 of 29)
Classical Approach
Two-Tailed

92. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (7 of 29)
Classical Approach
Left-Tailed

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11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (8 of 29)
Classical Approach
Right-Tailed

94. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (9 of 29)
Classical Approach
Step 4: Compare the critical value with the test statistic:

95. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (10 of 29)
P-value Approach
Step 3: Compute the test statistic

96. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (11 of 29)
P-value Approach
Use Table VII to determine the P-value using the smaller of n1 − 1 or n2 − 1 degrees of freedom.

97. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (12 of 29)
P-value Approach
Two-Tailed

98. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (13 of 29)
P-value Approach
Left-Tailed

99. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (14 of 29)
P-value Approach
Right-Tailed

100. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (15 of 29)
P-value Approach
Technology Step 3: Use a statistical spreadsheet or calculator with statistical capabilities to obtain the P-value. The directions for obtaining the P-value using the TI-83/84 Plus graphing calculator, Excel, MINITAB, and StatCrunch are in the Technology Step-by- Step in the text.

101. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (16 of 29)
P-value Approach
Step 4: If P-value < α, reject the null hypothesis.

102. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (17 of 29)
P-value Approach
Step 5: State the conclusion.

103. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (18 of 29)
These procedures are robust, which means that minor departures from normality will not adversely affect the results. However, if the data have outliers, the procedure should not be used.

104. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (19 of 29)
Parallel Example 1: Testing Hypotheses Regarding Two Means
A researcher wanted to know whether “state” quarters had a weight that is more than “traditional” quarters. He randomly selected 18 “state” quarters and 16 “traditional” quarters, weighed each of them and obtained the following data.

105. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (20 of 29)
STATE
(grams)
blank
TRADITIONAL
5.70
5.67
5.55
5.73
5.61
5.72
5.58
5.65
5.62
5.66
5.74
5.68
5.79
5.53
5.77
5.71
5.76

106. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (21 of 29)
Test the claim that “state” quarters have a mean weight that is more than “traditional” quarters at the α = 0.05 level of significance.
NOTE: A normal probability plot of “state” quarters indicates the population could be normal. A normal probability plot of “traditional” quarters indicates the population could be normal

107. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (22 of 29)
No outliers.

108. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (23 of 29)
Solution
Step 1: We want to determine whether state quarters weigh more than traditional quarters:
H0: μ1 = μ2 versus H1: μ1 > μ2
Step 2: The level of significance is α = 0.05.
Step 3: The test statistic is

109. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (24 of 29)
Solution: Classical Approach
This is a right-tailed test with α = Since n1 − 1 = 17 and n2 − 1 = 15, we will use 15 degrees of freedom. The corresponding critical value is t0.05 =

110. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (25 of 29)
Solution: Classical Approach
Step 4: Since the test statistic, t0 = 2.53 is greater than the critical value t.05 = 1.753, we reject the null hypothesis.

111. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (26 of 29)
Solution: P-value Approach
Because this is a right-tailed test, the P-value is the area under the t-distribution to the right of the test statistic t0 = That is, P-value = P(t > 2.53) ≈ 0.01.

112. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (27 of 29)
Solution: P-value Approach
Step 4: Since the P-value is less than the level of significance α = 0.05, we reject the null hypothesis.

113. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (28 of 29)
Solution
Step 5: There is sufficient evidence at the α = 0.05 level to conclude that the state quarters weigh more than the traditional quarters.

114. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Test Hypotheses Regarding the Difference of Two Independent Means (29 of 29)
NOTE: The degrees of freedom used to determine the critical value in the last example are conservative. Results that are more accurate can be obtained by using the following degrees of freedom:

115. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (1 of 8)
Constructing a (1 − α) • 100% Confidence Interval for the Difference of Two Means
A simple random sample of size n1 is taken from a population with unknown mean μ1 and unknown standard deviation σ1. Also, a simple random sample of size n2 is taken from a population with unknown mean μ2 and unknown standard deviation σ2. If the two populations are normally distributed or the sample sizes are sufficiently large (n1 ≥ 30 and n2 ≥ 30), a(1 − α) • 100% confidence interval about μ1 − μ2 is given by . . .

116. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (2 of 8)
Constructing a (1 − α) • 100% Confidence Interval for the Difference of Two Means

117. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (3 of 8)
Parallel Example 3: Constructing a Confidence Interval for the Difference of Two Means
Construct a 95% confidence interval about the difference between the population mean weight of a “state” quarter versus the population mean weight of a “traditional” quarter.

118. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (4 of 8)
Solution
We have already verified that the populations are approximately normal and that there are no outliers.

119. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (5 of 8)
Solution
Thus,

120. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (6 of 8)
Solution
We are 95% confident that the mean weight of the “state” quarters is between and ounces more than the mean weight of the “traditional” quarters. Since the confidence interval does not contain 0, we conclude that the “state” quarters weigh more than the “traditional” quarters.

121. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (7 of 8)
When the population variances are assumed to be equal, the pooled t-statistic can be used to test for a difference in means for two independent samples. The pooled t-statistic is computed by finding a weighted average of the sample variances and uses this average in the computation of the test statistic. The advantage of this test statistic is that it exactly follows Student’s t-distribution with n1 + n2 − 2 degrees of freedom. The disadvantage to this test statistic is that it requires that the population variances be equal.

122. 11. 3 Inference about Two Means: Independent Samples 11. 3
11.3 Inference about Two Means: Independent Samples Construct and Interpret Confidence Intervals Regarding the Difference of Two Independent Means (8 of 8)
CAUTION!
We would use the pooled two-sample t-test when the two samples come from populations that have the same variance. Pooling refers to finding a weighted average of the two sample variances from the independent samples. It is difficult to verify that two population variances might be equal based on sample data, so we will always use Welch’s t when comparing two means.

123. 11.4 Putting It Together: Which Method Do I Use? Learning Objectives
1. Determine the appropriate hypothesis test to perform

124. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (1 of 9)
What parameter is addressed in the hypothesis?
Proportion, p
σ or σ2
Mean, μ

125. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (2 of 9)
Proportion, p
Is the sampling Dependent or Independent?

126. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (3 of 9)
Proportion, p
Dependent samples:
Provided the samples are obtained randomly and the total number of observations where the outcomes differ is at least 10, use the normal distribution with

127. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (4 of 9)
Proportion, p
Independent samples:

128. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (5 of 9)
σ or σ2
Provided the data are normally distributed, use the F-distribution with

129. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (6 of 9)
Mean, μ
Is the sampling Dependent or Independent?

130. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (7 of 9)
Mean, μ
Independent samples:
Provided each sample size is greater than 30 or each population is normally distributed, use Student’s t-distribution

131. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (8 of 9)
Mean, μ
Dependent samples:
Provided each sample size is greater than 30 or the differences come from a population that is normally distributed, use Student’s t-distribution with n − 1 degrees of freedom with

132. 11. 4 Putting It Together: Which Method Do I Use. 11. 4
11.4 Putting It Together: Which Method Do I Use? Determine the Appropriate Hypothesis Test to Perform (9 of 9)