Statistics Chapter 9: Inferences Based on Two Samples: Confidence Intervals and Tests of Hypotheses

Where We’ve Been Made inferences based on confidence intervals and tests of hypotheses Studied confidence intervals and hypothesis tests for µ and p Selected the necessary sample size for a given margin of error McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

Where We’re Going Learn to use confidence intervals and hypothesis tests to compare two populations Learn how to use these tools to compare two population means, proportions and variances Select the necessary sample size for a given margin of error when comparing parameters from two populations McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.1: Identifying the Target Parameter µ1 - µ2 Mean difference; difference in averages Quantitative Data p1 - p2 Difference between proportions, percentages, fractions or rates; compare proportions Qualitative Data McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Point Estimators  → µ 1 - 2 → µ1 - µ2 To construct a confidence interval or conduct a hypothesis test, we need the standard deviation: Singe sample Two samples McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling The Sampling Distribution for (1 - 2) The mean of the sampling distribution is (µ1-µ2). If the two samples are independent, the standard deviation of the sampling distribution (the standard error) is The sampling distribution for (1 - 2) is approximately normal for large samples. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling The Sampling Distribution for (1 - 2) McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Large Sample Confidence Interval for (µ1 - µ2 ) McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Two samples concerning retention scores for first-year students at private and public institutions were obtained from the Department of Education’s data base to see if there was a significant difference in the two types of colleges. Private Colleges Public Universities n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17 n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 What does a 95% confidence interval tell us about retention scores? Source: National Center for Education Statistics McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Private Colleges Public Universities n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17 n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Private Colleges Public Universities n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17 n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 Since 0 is not in the confidence interval, the difference in the sample means appears to indicate a real difference in retention. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling One-Tailed Test Two-Tailed Test H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) < D0 (> D0) Rejection region: z < -z (> z ) Test Statistic: where H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) ≠ D0 Rejection region: |z| > z /2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Conditions Required for Valid Large-Sample Inferences about (µ1 - µ2) 1. The two samples are randomly and independently selected from the target populations. 2. The sample sizes are both ≥ 30. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Let’s go back to the retention data and test the hypothesis that there is no significant difference in retention at privates and publics. Private Colleges Public Universities n: 71 Mean: 78.17 Standard Deviation: 9.55 Variance: 91.17 n: 32 Mean: 84 Standard Deviation: 9.88 Variance: 97.64 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Test statistic: Reject the null hypothesis: McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling For small samples, the t-distribution can be used with a pooled sample estimator of 2, sp2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Small Sample Confidence Interval for (µ1 - µ2 ) The value of t is based on (n1 + n2 -2) degrees of freedom. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling One-Tailed Test Two-Tailed Test H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) < D0 (> D0) Rejection region: t < -t (> t ) Test Statistic: H0: (µ1 - µ2) = D0 Ha: (µ1 - µ2) ≠ D0 Rejection region: |t| > t /2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Conditions Required for Valid Small-Sample Inferences about (µ1 - µ2) 1. The two samples are randomly and independently selected from the target populations. 2. Both sampled populations have distributions that are approximately normal. 3. The population variances are equal. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling Does class time affect performance? The test performance of students in two sections of international trade, meeting at different times, were compared. 8:00 a.m. Class Mean: 78 Standard Deviation: 14 Variance: 196 n: 21 9:30 a.m. Class Mean: 82 Standard Deviation: 17 Variance: 289 n: 21 With = .05, test H0 : µ1 = µ2 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 78 Variance: 196 n: 21 9:30 a.m. Class Mean: 82 Variance: 289 n: 21 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 78 Variance: 196 n: 21 9:30 a.m. Class Mean: 82 Variance: 289 n: 21 With df = 21 + 21 – 2 = 40, t /2 = t.025 = 2.021. Since out test statistic t = -.812. |t| < t.025. Do not reject the null hypothesis McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.2: Comparing Two Population Means: Independent Sampling 8:00 a.m. Class Mean: 72 Variance: 154 n: 13 9:30 a.m. Class Mean: 86 Variance: 163 n: 21 Your text book uses the following formula when the sample sizes are DIFFERENT, but you do NOT need to use this in Math 201. Just use the approach of the preceding slides even when McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments Suppose ten pairs of puppies were housetrained using two different methods: one puppy from each pair was paper-trained, with the paper gradually moved outside, and the other was taken out every three hours and twenty minutes after each meal. The number of days until the puppies were considered housetrained (three days straight without an accident) were compared. Nine of the ten paper-trained dogs took longer than the other paired dog to complete training, with the average difference equal to 4 days, with a standard deviation of 3 days. What is a 90% confidence interval on the difference in successful training? McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments Since 0 is not in the interval, one program does seem to work more effectively. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference: $1.75 Standard Deviation: $10.35 Test at the 5% level of significance that the difference in the two stores is zero. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference: $1.75 Standard Deviation: $10.35 = .05 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.3: Comparing Two Population Means: Paired Difference Experiments Suppose 150 items were priced at two online stores, “cport” and “warriorwoman.” Mean difference: $1.75 Standard Deviation: $10.35 = .05 The critical value z.025 is 1.96, so we would reject this null hypothesis. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling A group of men and women were asked their opinions about an important issue (!): Is the new comedy series (for TV) funny? The results are as follow: Men Women Yes 290 200 No 50 n 340 250 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling Calculate a 95% confidence interval on the difference in the opinions of men and women. Men Women p .85 .80 q .15 .20 n 340 250 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling Calculate a 95% confidence interval on the difference in the opinions of men and women. Since 0 is in the confidence interval, we cannot rule out the possibility that both genders find the series equally funny. Men Women p .85 .80 q .15 .20 n 340 250 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling Randy Stinchfield of the University of Minnesota studied the gambling activities of public school students in 1992 and 1998 (Journal of Gambling Studies, Winter 2001). His results are reported below: Do these results represent a statistically significant difference at the = .01 level? 1992 1998 Survey n 21,484 23,199 Number who gambled 4.684 5,313 Proportion who gambled .218 .229 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling 1992 1998 Survey n 21,484 23,199 Number who gambled 4.684 5,313 Proportion who gambled .218 .229 McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling 1992 1998 Survey n 21,484 23,199 Number who gambled 4.684 5,313 Proportion who gambled .218 .229 Since the computed value of z, -2.786, is of greater magnitude than the critical value, 2.576, we can reject the null hypothesis at the = .01 level. McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples

9.4: Comparing Two Population Proportions: Independent Sampling For valid inferences The two samples must be independent The two sample sizes must be large: McClave, Statistics, 11th ed. Chapter 9: Inferences Based on Two Samples