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Chapter 9. Comparing Two populations. Section 9.1 and 9.2. Comparing Means. Example 1. A biologist who studies spiders is interested in comparing the lengths of female and male green lynx spiders. Assume that the length X of the male spider is N(X,2X). You have a sample of size n from this population. Assume that the length Y of the female spider is N(Y,2Y). You have a sample of size m from this population. Assume that the X data is independent of the Y data. Male lengths: 6.3, 6.3, 4.6, 6.5, 5.8 (m=5, x_bar =5.9, sX=0.75) 4 6 Female lengths: 7.9, 10.3, 9.4, 11.2 (n=4, y_bar =9.7, sY=1.4) 5 8 6 335 9 7 8 4 9 3 10 2 11 Estimate X-Y by constructing a 100(1-)% confidence interval. Example 2. A manufacturing company has two different saws used for cutting columns. On average, they suspect one saw (saw X) may be cutting columns shorter than the other (saw Y). Does the following data provide significant evidence in support of this suspicion? X: 8.02 8.10 8.04 8.04 8.00 8.11 8.07 8.02 8.04 (m=9, x =8.05, sX=0.037) Y: 8.04 8.04 8.10 8.06 8.08 8.10 8.07 8.08 8.06 (n=9, y =8.07, sX=0.022) The following box plots confirm that the data supports the claim, but is it significant? 8.10 Lengths 8.05 8.00 X Y Code What is a point estimator for 1-2? What is the sampling distribution? Give the standardized estimator. Case Confidence Interval Test Statistic If both populations are assumed to have the same standard deviation, how does the variance of the sampling distribution simplify? What is a reasonable point estimator for the common sigma? What are the degrees of freedom? Give an associated statistic that has a t distribution If you do not assume the standard deviations of both populations are the same, how do you estimate the variance of the sampling distribution? Does this lead to a test statistic with a t distribution? Case Confidence Interval Test Statistic When to use Computations for example 1. Computations for example 2. (Use the critical region approach.) What is the P-value for the data in example 2? Example 3. An experiment to compare the tension bond strength of polymer latex modified mortar to unmodified mortar resulted in the following data m 40, x 18.12, n 32, y 16.87 a. Assuming that 1 = 1.6 and 2 = 1.4, is this convincing evidence that the modified mortar gives a higher average tension bond strength? b. Compute the probability of type II error when the true average difference is 1? c. Suppose the 40 specimens of modified mortar have already been collected. How many unmodified specimens should be collected so that the power of the 0.05 level test to detect a true average difference of 1 is 90%? d. How would the conclusion of part a change if 1.6 and 1.4 were sample standard deviations instead? Section 9.3 Paired Data. Example 4. Makers of generic drugs must show that they do not differ significantly from the reference drug that they imitate. One aspect they might differ in is their extent of absorption in the blood. a. To study this, 10 subjects are randomly divided into two groups. The first group is given only the reference drug. The second group is given only the generic drug. The absorption of the given drug was measured. Reference Drug Generic Drug 995 2803 2166 2019 3131 1651 2012 1156 1775 2107 Estimate with 95% confidence the difference in mean absorption rates for the two drugs. b. The same thing is being studied, but this time 10 subjects are randomly divided into two groups. The first group was given the generic drug first, the second group was given the reference drug first. In all cases, a washout period separated the two drugs so that the first had disappeared from the blood before the subject took the second. The absorption of each drug was measured. Subject Reference Drug Generic Drug 1 1022 1284 2 1339 1930 3 2463 2120 4 2779 1613 5 2256 3052 6 1438 2549 7 1833 1310 8 3852 2254 9 1262 1964 10 4108 1755 Use the new data to estimate with 95% confidence the difference in mean absorption rates for the two drugs. Comments on Matched Pairs In our example, the same subject was subjected to both treatments. When this cannot be done, paired data can be obtained by matching subjects. For example, divide 10 subjects into 5 pairs according to blood pressure. In each pair, one person is given reference drug and one generic drug. In a paired design you have half the degrees of freedom as in a two sample design. Pairing is still useful when enough variability in the response to the two treatments is controlled by the pairing to overcome the loss of degrees of freedom. Section 9.4. Comparing proportions. Example 5. Mailing letters. Summarize data by giving sample proportion of all letters that would be mailed. Construct a 95% confidence interval for the proportion of all Hope students that would mail such a letter (consider the people in this class to be representative of the population). Notations. What is a point estimator for p1- p 2? What is the sampling distribution? Confidence Interval Test Statistic Example 5b. Estimate the difference in the two population proportions of interest. Example 5c. Is there significant evidence that p1> p 2?. Suppose p1 = 0.75 and p2 = 0.35. Using equal sample sizes of 12, what is the power of the test? What sample size would be required so that the 0.05 level test detects the above specified difference with a power of 90%? Homework assignment for Chapter 9. Section 9.1 1, 4, 5, 7, 10 (Hint on number 10: You first have to calculate s for each sample The standard errors are 0.3 and 0.2 respectively, but the standard errors are s divided by the square root of n). Section 9.2 19, 20, 25, 27, 28 Section 9.3 37, 41, 45 Section 9.4 48, 49, 50, 51