Does the plot of the data suggest any violation of the conditions necessary to use the chi- Place bounds on the p-value of the test in Exercise 7.7. Engin.

a. Does the plot indicate any violation of the conditions underlying the use of the

chi-square procedures for constructing confidence intervals or testing hypotheses about ?

b. Is there sufficient evidence that the standard deviation in rebound coefficient for the

batch of balls is less than 2?

c. Estimate the standard deviation of the rebound coefficients using a 95 confidence

interval. s 81 82 83 84 85 86 87 88 89 90 Rebound coef ficient .999 .99 .95 .80 .50 .20 .05 .01 .001 81 82 83 84 85 86 87 88 89 90 Rebound coefficient Probability 7.10 Use the results of the simulation study, summarized in Table 7.2, to answer the following questions.

a. Which of skewness or heavy-tailedness appears to have the strongest effect on the

chi-square tests?

b. For a given population distribution, does increasing the sample size yield values

more nearly equal to the nominal value of .05? Justify your answer and provide rea- sons why this may occur.

c. For the short-tailed distribution Uniform, the actual probability of Type I error is

smaller than the specified value of .05. Provide both a negative and positive impact on the chi-square test of having a decrease in the specified value of .

7.3 Estimation and Tests for Comparing Two Population Variances

7.11 Find the value of F that locates an area in the upper tail of the F distribution; that is, find for the following specifications:

a. b. a ⫽ .025, df

1 ⫽ 15, df 2 ⫽ 15 a ⫽ .05, df 1 ⫽ 5, df 2 ⫽ 15 F a a a a

c. d.

e. 7.12 Find the value of F that locates an area in the lower tail of the F distribution; that is, find for the following specifications:

a. b.

c. d.

e.

7.13 Find approximate values for for the following specifications:

a. b.

c. d.

e.

7.14 Random samples of n

1 ⫽ 15 and n 2 ⫽ 10 were selected from populations 1 and 2, respec- tively. The corresponding sample standard deviations were s 1 ⫽ 5.3 and s 2 ⫽ 8.8.

a. Do the data provide sufficient evidence to indicate a difference in

and ?

b. Place a 95 confidence interval on the ratio of the variances .

c. What assumptions have you made concerning the data and populations when making

your calculations in parts a and b? Engin. 7.15 A soft-drink firm is evaluating an investment in a new type of canning machine. The com- pany has already determined that it will be able to fill more cans per day for the same cost if the new machines are installed. However, it must determine the variability of fills using the new machines, and wants the variability from the new machines to be equal to or smaller than that currently obtained using the old machines. A study is designed in which random samples of 61 cans are selected from the output of both types of machines and the amount of fill in ounces is determined. The data are summarized in the following table and boxplots. Summary Data for Canning Experiment Machine Type Sample Size Mean Standard Deviation Old 61 12.284 .231 New 61 12.197 .162 s 2 1 兾s 2 2 s 2 s 1 a ⫽ .05 a ⫽ .005, df 1 ⫽ 90, df 2 ⫽ 75 a ⫽ .001, df 1 ⫽ 35, df 2 ⫽ 35 a ⫽ .01, df 1 ⫽ 50, df 2 ⫽ 12 a ⫽ .025, df 1 ⫽ 35, df 2 ⫽ 15 a ⫽ .05, df 1 ⫽ 14, df 2 ⫽ 19 F a a ⫽ .005, df 1 ⫽ 8, df 2 ⫽ 13 a ⫽ .001, df 1 ⫽ 15, df 2 ⫽ 5 a ⫽ .01, df 1 ⫽ 10, df 2 ⫽ 12 a ⫽ .025, df 1 ⫽ 15, df 2 ⫽ 15 a ⫽ .05, df 1 ⫽ 5, df 2 ⫽ 15 F 1⫺ a a a ⫽ .005, df 1 ⫽ 8, df 2 ⫽ 13 a ⫽ .001, df 1 ⫽ 15, df 2 ⫽ 5 a ⫽ .01, df 1 ⫽ 10, df 2 ⫽ 12 12.8 12.3 11.8 Old machine New machine Boxplots of old machine and new machine means are indicated by solid circles