10-3 INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN

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it. Formulate and test an appropriate hypothesis, using 0.05. (c) What is the P-value for the test you conducted in part (b) 10-8. Consider the situation described in Exercise 10-4. What sample size would be required in each population if we wanted the error in estimating the difference in mean burning rates to be less than 4 centimeters per second with 99% con dence 10-9. Consider the road octane test situation described in Exercise 10-7. What sample size would be required in each population if we wanted to be 95% con dent that the error in estimating the difference in mean road octane number is less than 1 10-10. A polymer is manufactured in a batch chemical process. Viscosity measurements are normally made on each batch, and long experience with the process has indicated that the variability in the process is fairly stable with 20. Fifteen batch viscosity measurements are given as follows: 724, 718, 776, 760, 745, 759, 795, 756, 742, 740, 761, 749, 739, 747, 742. A process change is made which involves switching the type of catalyst used in the process. Following the process change, eight batch viscosity measurements are taken: 735, 775, 729, 755, 783, 760, 738, 780. Assume that process variability is unaffected by the catalyst change. Find a 90% con dence interval on the difference in mean batch viscosity resulting from the process change. 10-11. The concentration of active ingredient in a liquid laundry detergent is thought to be affected by the type of catalyst used in the process. The standard deviation of active concentration is known to be 3 grams per liter, regardless of the catalyst type. Ten observations on concentration are taken with each catalyst, and the data follow: Catalyst 1: 57.9, 66.2, 65.4, 65.4, 65.2, 62.6, 67.6, 63.7, 67.2, 71.0 Catalyst 2: 66.4, 71.7, 70.3, 69.3, 64.8, 69.6, 68.6, 69.4, 65.3, 68.8

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(a) Find a 95% con dence interval on the difference in mean active concentrations for the two catalysts. (b) Is there any evidence to indicate that the mean active concentrations depend on the choice of catalyst Base your answer on the results of part (a). 10-12. Consider the polymer batch viscosity data in Exercise 10-10. If the difference in mean batch viscosity is 10 or less, the manufacturer would like to detect it with a high probability. (a) Formulate and test an appropriate hypothesis using 0.10. What are your conclusions (b) Calculate the P-value for this test. (c) Compare the results of parts (a) and (b) to the length of the 90% con dence interval obtained in Exercise 10-10 and discuss your ndings. 10-13. For the laundry detergent problem in Exercise 10-11, test the hypothesis that the mean active concentrations are the same for both types of catalyst. Use 0.05. What is the P-value for this test Compare your answer to that found in part (b) of Exercise 10-11, and comment on why they are the same or different. 10-14. Reconsider the laundry detergent problem in Exercise 10-11. Suppose that the true mean difference in active concentration is 5 grams per liter. What is the power of the test to detect this difference if 0.05 If this difference is really important, do you consider the sample sizes used by the experimenter to be adequate 10-15. Consider the polymer viscosity data in Exercise 1010. Does the assumption of normality seem reasonable for both samples 10-16. Consider the concentration data in Exercise 10-11. Does the assumption of normality seem reasonable

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10-3 INFERENCE FOR THE DIFFERENCE IN MEANS OF TWO NORMAL DISTRIBUTIONS, VARIANCES UNKNOWN

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We now extend the results of the previous section to the difference in means of the two distributions in Fig. 10-1 when the variances of both distributions 2 and 2 are unknown. If the sam1 2 ple sizes n1 and n2 exceed 40, the normal distribution procedures in Section 10-2 could be used. However, when small samples are taken, we will assume that the populations are normally distributed and base our hypotheses tests and con dence intervals on the t distribution. This nicely parallels the case of inference on the mean of a single sample with unknown variance.

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10-3.1

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