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Chapter. 11. Inferences on Two Samples. Section. 11.1. Inference about Two Means: Dependent Samples. Objectives. Distinguish between independent and dependent sampling Test hypotheses regarding matched-pairs data

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  1. Chapter 11 Inferences on Two Samples

  2. Section 11.1 Inference about Two Means: Dependent Samples

  3. Objectives • Distinguish between independent and dependent sampling • Test hypotheses regarding matched-pairs data • Construct and interpret confidence intervals about the population mean difference of matched-pairs data

  4. Objective 1 • Distinguish between Independent and Dependent Sampling

  5. 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. Dependent samples are often referred to as matched-pairs samples.

  6. 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.

  7. 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.

  8. Objective 2 • Test Hypotheses Regarding Matched-Pairs Data

  9. “In Other Words” Statistical inference methods on matched-pairs data use the same methods as inference on a single population mean with  unknown, except that the differences are analyzed.

  10. Testing Hypotheses Regarding the Difference of Two Means Using a Matched-Pairs Design To test hypotheses regarding the mean difference of matched-pairs data, the following must be satisfied: • 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).

  11. Step 1:Determine the null and alternative hypotheses. The hypotheses can be structured in one of three ways, where d is the population mean difference of the matched-pairs data.

  12. Step 2:Select a level of significance, , based on the seriousness of making a Type I error.

  13. Step 3:Compute the test statistic which approximately follows Student’s t-distribution with n-1 degrees of freedom. The values of and sd are the mean and standard deviation of the differenced data.

  14. Classical Approach Step 4:Use Table VI to determine the critical value using n-1 degrees of freedom.

  15. (critical value) Classical Approach Two-Tailed

  16. (critical value) Classical Approach Left-Tailed

  17. (critical value) Classical Approach Right-Tailed

  18. Classical Approach Step 5:Compare the critical value with the test statistic:

  19. P-Value Approach Step 4:Use Table VI to determine the P-value using n-1 degrees of freedom.

  20. P-Value Approach Two-Tailed

  21. P-Value Approach Left-Tailed

  22. P-Value Approach Right-Tailed

  23. P-Value Approach Step 5:If P-value < , reject the null hypothesis.

  24. Step 6:State the conclusion.

  25. 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.

  26. 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 =0.05 level of significance.

  27. 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. The differences (Hampton - La Quinta) are: 24 53 100 40 -10 71 77 70 39 50 with = 51.4 and sd = 30.8336.

  28. Solution No violation of normality assumption.

  29. Solution No outliers.

  30. Solution Step 1: We want to determine if the prices differ: H0: d = 0 versus H1: d  0 Step 2: The level of significance is =0.05. Step 3: The test statistic is

  31. Solution: Classical Approach Step 4: This is a two-tailed test so the critical values at the =0.05 level of significance with n-1=10-1=9 degrees of freedom are -t0.025=-2.262 and t0.025=2.262.

  32. Solution: Classical Approach Step 5: Since the test statistic, t0=5.27 is greater than the critical value t.025=2.262, we reject the null hypothesis.

  33. Solution: P-Value Approach Step 4: 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(0.00026)=0.00052 (using technology). Approximately 5 samples in 10,000 will yield results as extreme as we obtained if the null hypothesis is true.

  34. Solution: P-Value Approach Step 5: Since the P-value is less than the level of significance =0.05,we reject the null hypothesis.

  35. Solution Step 6: There is sufficient evidence to conclude that Hampton Inn hotels and La Quinta hotels are priced differently at the =0.05 level of significance.

  36. Objective 3 • Construct and Interpret Confidence Intervals for the Population Mean Difference of Matched-Pairs Data

  37. Confidence Interval for Matched-Pairs Data A (1-)100% confidence interval for d is given by Lower bound: Upper bound: The critical value t/2 is determined using n-1 degrees of freedom.

  38. 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.

  39. 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.

  40. Solution • We have already verified that the differenced data come from a population that is approximately normal with no outliers. • Recall = 51.4 and sd = 30.8336. • From Table VI with  = 0.10 and 9 degrees of freedom, we find t/2 = 1.833.

  41. Solution Thus, • Lower bound = • Upper bound = We are 90% confident that the mean difference in hotel room price for Ramada Inn versus La Quinta Inn is between $33.53 and $69.27.

  42. Section 11.2 Inference about Two Means: Independent Samples

  43. Objectives • Test hypotheses regarding the difference of two independent means • Construct and interpret confidence intervals regarding the difference of two independent means

  44. 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 1 and unknown standard deviation 1. In addition, a simple random sample of size n2 is taken from a population with unknown mean 2 and unknown standard deviation 2. If the two populations are normally distributed or the sample sizes are sufficiently large (n1 ≥ 30, n2 ≥ 30) , then approximately follows Student’s t-distribution with the smaller of n1-1 or n2-1 degrees of freedom where is the sample mean and si is the sample standard deviation from population i.

  45. Objective 1 • Test Hypotheses Regarding the Difference of Two Independent Means

  46. Testing Hypotheses Regarding the Difference of Two Means To test hypotheses regarding two population means, 1 and 2, 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).

  47. Step 1:Determine the null and alternative hypotheses. The hypotheses are structured in one of three ways:

  48. Step 2:Select a level of significance, , based on the seriousness of making a Type I error.

  49. Step 3:Compute the test statistic which approximately follows Student’s t- distribution.

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