Estimate the mean percentage of SiO Evaluate whether the percentage of SiO Use the following plot to determine if the procedures you used in parts a and b

5.78 Refer to Exercise 5.77. a. Estimate the median percentage of SiO

2 using a 95 confidence interval.

b. Evaluate whether the median percentage of SiO

2 is different from 6.2 using an a ⫽ .05 test of hypothesis.

5.79 Refer to Exercise 5.77. Generate 1,000 bootstrap samples from the 20 SiO

2 percentages.

a. Construct a 95 bootstrap confidence interval on the mean SiO

2 percentage. Compare this interval to the interval obtained in Exercise 5.77a.

b. Obtain the bootstrap p-value for testing whether the mean percentage of SiO

2 differs from 6.2. Compare this value to the p-value for the test in Exercise 5.77b.

c. Why is there such a good agreement between the t-based and bootstrap values in

parts a and b? Med. 5.80 A medical team wants to evaluate the effectiveness of a new drug that has been proposed for people with high intraocular pressure IOP. Prior to running a full-scale clinical trial of the drug, a pilot test was run using 10 patients with high IOP values. The n ⫽ 10 patients had a mean decrease in IOP of ⫽ 15.2 mm Hg with a standard deviation of the 10 IOPs equal to s ⫽ 9.8 mm Hg after 15 weeks of using the drug. Determine the appropriate sample size for an a ⫽ .01 test to have at most a .10 probability of a failing to detect at least a 4 mm Hg decrease in the mean IOP. y 2 4 6 8 10 12 SiO 2 Evaluation of normality 5 1

10 20

Percent 30 40 50 60 70 80 90 95 99 P -Value 0.100 RJ 0.975 N 20 StDev 2.503 Mean 5.673 290 CHAPTER 6 Inferences Comparing Two Population Central Values 6.1 Introduction and Abstract of Research Study 6.2 Inferences about M 1 ⴚ M 2 : Independent Samples 6.3 A Nonparametric Alternative: The Wilcoxon Rank Sum Test 6.4 Inferences about M 1 ⴚ M 2 : Paired Data 6.5 A Nonparametric Alternative: The Wilcoxon Signed-Rank Test 6.6 Choosing Sample Sizes for Inferences about M 1 ⴚ M 2 6.7 Research Study: Effects of Oil Spill on Plant Growth 6.8 Summary and Key Formulas 6.9 Exercises

6.1 Introduction and Abstract of Research Study

The inferences we have made so far have concerned a parameter from a single population. Quite often we are faced with an inference involving a comparison of parameters from different populations. We might wish to compare the mean corn crop yield for two different varieties of corn, the mean annual income for two ethnic groups, the mean nitrogen content of two different lakes, or the mean length of time between administration and eventual relief for two different antivertigo drugs. In many sampling situations, we will select independent random samples from two populations to compare the populations’ parameters. The statistics used to make these inferences will, in many cases, be the difference between the corre- sponding sample statistics. Suppose we select independent random samples of n 1 observations from one population and n 2 observations from a second population. We will use the difference between the sample means, , to make an infer- ence about the difference between the population means, . The following theorem will help in finding the sampling distribution for the difference between sample statistics computed from independent random samples. m 1 ⫺ m 2 y 1 ⫺ y 2