In Exercise 64, we found a confidence interval for the proportion of American consumers who believe they own an American-made television set. How many sample observations would be needed to be sure that a 95% confidence interval for the population proportion extends no more than .05 on each side of the sample proportion?

1. A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weights is known to be normal, with standard devia- tion 4 ounce. A random sample of sixteen boxes yielded a sample mean weight of 15.84. ounces. Test at the 10% significance level the null hypothesis that the population mean weight is at least 16 ounces. 2. A company which receives shipments of batteries tests a random sample of nine of them before agreeing to take a shipment. The company is concemed that the true mean lifetime for all batteries in the shipment should be at least 50 hours. From past experience, it is safe to conclude that the population distribution of lifetimes is normal, with standard deviation 3 hours. For one particular shipment, the mean lifetime for a sample of nine batteries was 48.2 hours. Test at the 10% level the null hypothesis that the population mean lifetime is at least 50 hours. 3. A pharmaceutical manufacturer is concerned about the impurity concentration in pills, and it is anxious that this concentration not exceed 3%. It is known that from a particular pro- duction run, impurity concentrations follow a normal distribution with standard deviation 4%. A random sample of sixty-four pills from a production run was checked, and the sample mean impurity concentration was found to be 3.07%.

(a) Test at the 5% level the null hypothesis that the population mean impurity concentra- tion is 3% against the alternative that it is more than 3%.

(b) Find the p-value for this test.

(c) Suppose that the alternative hypothesis had been two-sided rather than one-sided (with null hypothesis Hp 3). State, without doing the calculations, whether the p-value of the test would be higher than, lower than, or the same as that found in (b). Sketch a graph to illustrate your reasoning.

(d) In the context of this problem, explain why a one-sided alternative hypothesis is more appropriate than a two-sided altemative. 4. A manufacturer claims that through the use of a fuel additive, automobiles should achieve on average an additional 3 miles per gallon of gas. A random sample of 100 automobiles was used to evaluate this product. The sample mean increase in miles per gallon achieved was 2.4 and the sample standard deviation was 1.8 miles per gallon. Test the null hypoth esis that the population mean is at least 3 miles per gallon. Find the p-value of this test, and interpret your findings. 5. A random sample of 1,562 undergraduates enrolled in marketing courses was asked to re- spond on a scale from one (strongly disagree) to seven (strongly agree) to the proposition: "Advertising helps raise our standard of living." The sample mean response was 4.27 and the sample standard deviation was 1.32. Test at the 1% level, against a two-sided alterna- tive, the null hypothesis that the population mean is 4. 6. A random sample of 76 percentage changes in promised pension benefits of single employer plans after the establishment of the Pension Benefit Guarantee Corporation was observed. The sample mean percentage change was .078 and the sample standard devia- tion was .201. Find and interpret the p-value of a test of the null hypothesis that the popu- lation mean percentage change is 0, against a two-sided alternative.

7. A random sample of 172 accounting students was asked to rate on a scale from one (not important) to five (extremely important) starting salary as a job characteristic." The sample mean rating was 3.31 and the sample standard deviation was .70. Test at the 1% signifi- cance level the null hypothesis that the population mean rating is at most 3.0 against the alternative that it is bigger than 3.0. 8. A random sample of 170 people was provided with a prediction problem." Each sample member was given, in two ways, the task of projecting the next value of an economic vari- able. The previous twenty values were presented both as numbers and as points on a graph. Subjects were asked to predict the next value, as a number and as a point on the graph. The absolute prediction errors were measured. The sample then consisted of 170. differences in absolute forecast errors (numerical minus graphical). The sample mean of these differences was -2.91 and the sample standard deviation was 11.33. Find and inter-1 pret the p-value of a test of the null hypothesis that the population mean difference is 0. against the alternative that it is negative. (The alternative can be viewed as the hypothesis that, in the aggregate, people are more successful at graphical than numerical prediction.) 9. The accounts of a corporation show that on average accounts receivable are $125.32. An auditor checked a random sample of sixteen of these accounts. The sample mean was $131.78 and the sample standard deviation was $25.41. Assume that the population distri- bution is normal. Test at the 5% significance level against a two-sided alternative the null hypothesis that the population mean is $125.32.