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Statistical Inferences Based on Two Samples

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Statistical Inferences Based on Two Samples

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    1. Statistical Inferences Based on Two Samples 9.2 Comparing Two Population Means Using Small Independent Samples and Assuming Sigmas are Unknown 9.3 Paired Difference Experiments

    2. Sampling Distribution of

    3. Sampling Distribution of (Continued)

    4. Large Sample Confidence Interval, Difference in Mean

    5. 9.2 Comparing Two Population Means Using Independent Samples with Sigmas Unknown

    6. Tests about Differences in Means When Variances are Equal

    7. Hypothesis Test and Confidence Interval Example Exercise 9.21, pg. 369 What are we given? n1 = 22; s1 = 225; xbar1 = 1500; n2 = 22; s2 = 251; xbar2 = 1300; ? = .05; First assume equal population variances Step 1, establish hypotheses H0: ?1 - ?2 = 0 vs. Ha: ?1 - ?2 > 0 Step 2, set significance level. a = .05 (given) Step 3, compute the test statistic, but first the pooled variance

    8. Hypothesis Test and Confidence Interval Example; Exercise 9.21, pg. 369 Step 4a, determine the rejection point, t.05,42 ˜ 1.684 Step 4b, estimate the p-value. Using df = 42, t-table gives P(T > 3.307) = .001 and P(T > 2.704) =.005 Since 2.704 < (t = 2.78) < 3.307, p-value is between 0.001 and 0.005 Step 5, decision; reject Ho since (a) test statistic, t (2.78) > rejection point (1.684) or (b) p-value (between .001 & .005) < ? = .05

    10. Hypothesis Test Example Step 6, conclusion within context: there is very strong evidence that type A training results in higher mean weekly sales than does type training.

    11. MegaStat Output for Example

    13. 9.3 Paired Difference Experiments Before, we drew random samples from two different populations Now, have two different processes (or methods) Draw one random sample of units and use those units to obtain the results of each process For instance, use the same individuals for the results from one process vs. the results from the other process E.g., use the same individuals to compare “before” and “after” treatments By using the same individuals, we eliminate any differences in the individuals themselves and just compare the results from the two processes

    14. Paired Difference Experiments Continued Let md be the mean of population of paired differences md = m1 – m2 , where m1 is the mean of population 1 and m2 is the mean of population 2 Let and sd be the mean and standard deviation of a sample of paired differences that has been randomly selected from the population is the mean of the differences between pairs of values from both samples

    15. t-Based Confidence Interval for Paired Differences in Means

    16. Test Statistic for Paired Differences The test statistic is D0 = m1 – m2 is the claimed or actual difference between the population means D0 varies depending on the situation Often D0 = 0, and the null means that there is no difference between the population means The sampling distribution of this statistic is a t distribution with (n – 1) degrees of freedom

    17. Paired Differences Testing Rules

    18. Example on Inferences with Paired Samples Exercise 9.32, pg. 377

    19. Example on Inferences with Paired Samples Exercise 9.32, pg. 377 Key sample information n = 10; s = 3.02; xbar = 4.0; ? = .10 - .001; Step 1, establish hypotheses H0: ?d = ?pst - ?pre = 0 vs. Ha: ?d = ?pst - ?pre > 0 Step 2, set significance level. a = .05 (mid-range) Step 3, compute the test statistic

    20. Hypothesis Test and Confidence Interval Example; Exercise 9.21, pg. 369 Step 4a, determine the rejection point, t.05,9 = 1.833 Step 4b, estimate the p-value. Using df = 9, t-table gives P(T > 3.25) = .005 and P(T > 4.297) =.001 Since 3.25 < (t = 4.19) < 4.297, p-value is between 0.001 and 0.005 Step 5, decision; reject Ho since (a) test statistic, t (4.19) > rejection point (1.833) or (b) p-value (between .001 & .005) < ? = .05. Note, we would F.T.R. Ho only at ? = .001 of given range

    22. Hypothesis Test Example Step 6, conclusion within context: there is very strong evidence that post-exposure attitude scores are higher on average than pre-exposure attitude scores. In other words, advertisement appears to increase mean attitude scores.

    24. MegaStat Output for Paired Diff. Example

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