18 Problems in Statistics

1.
a.11 10 8 4 6 7 11 6 11 7
b.13 23 15 13 18 13 15 14 20 20 18 17 20 13

i. Find the range, mean, variance, and standard deviation of the population data set.

ii. Without calculating, which data set has the greatest sample standard deviation? Which has the least sample standard deviation? Explain your reasoning.

iii. How are the data sets the same? How do they differ?

2. Water Consumption The number of gallons of water consumed per day by a small village are listed. Make a frequency distribution (using five classes) for the data set. Then approximate the population mean and the population standard deviation of the data set.

167 180 192 173 145 151 174 175 178 160
195 224 244 146 162 146 177 163 149 188

3. Use frequency distribution formulas to approximate the sample mean and standard deviation of the following data set

108 139 120 123 120 132 123 131 131 157 150 124 111
101 135 119 116 117 127 128 139 119 118 114 127 Â

4. Weekly salaries (in dollars) for a sample of registered nurses are listed.

774 446 1019 795 908 667 444 960

i. Find the mean, the median, and the mode of the salaries. Which best describes a typical salary?
ii. Find the range, variance, and standard deviation of the data set. Interpret the results in the context of the real-life setting.

Confidence Intervals for the Mean (Small Samples)

5. Suppose you incorrectly used the normal distribution of find the maximum error of estimate for the given values of c, s, and n.
c = 0.95, s = 5, n = 16

i.Find the value of E using the normal distribution.
ii.Find the correct value using a t-distribution. Compare the results.

6.Suppose you incorrectly used the normal distribution of find the maximum error of estimate for the given values of c, s, and n.
c = 0.99, c = 3, n = 6

i. Find the value of E using the normal distribution.
ii. Find the correct value using a t-distribution. Compare the results.

7. You want to estimate the mean repair cost for dishwashers. The estimate must be within $10 of the population mean. Determine the required sample size to construct a 99% confidence interval for the population mean. Assume the population standard deviation is $22.50.

8. The following data set represents the repair costs (in dollars) for a random sample of 30 dishwashers.

41.82 52.81 57.80 68.16 73.48 78.88 88.13 88.79
90.07 90.35 91.68 91.72 93.01 95.21 95.34 96.50
100.05 101.32 103.59 104.19 105.62 111.32 117.14 118.42
118.77 119.01 120.70 140.52 141.84 147.06

i. Assume the population of dishwasher repair costs is normally distributed.
ii. Construct a 95% confidence interval for the population variance.
iii. Construct a 95% confidence interval for the population standard deviation.

9. An auto maker estimates that the mean gas mileage of its luxury sedan is at least 25 miles per gallon. A random sample of eight such cars had a mean of 23 miles per gallon and a standard deviation of 5 miles per gallon. At a = 0.05, can you reject the auto maker's claim that the mean gas mileage of its luxury sedan is at least 25 miles per gallon? Assume the population is normally distributed.

1. State the claim mathematically. Identify H0 and Ha.
2. Determine when a type I or type II error occurs.
3. Determine whether the hypothesis test is a one-tailed or a two-tailed test and whether to use a z-test, a t-test. Explain.
4. If necessary, find the critical value(s) and identify the rejection region(s).
5. Find the appropriate test statistic. If necessary, find the P-value.
6. Decide whether to reject or fail to reject the null hypothesis. Then interpret the decision in the context of the original claim.

10. A state school administrator says that the standard deviation of SAT verbal test scores is 105. A random sample of 14 SAT verbal test scores has a standard deviation of 113. At a = 0.01, test the administrator's claim. What can you conclude? Assume the population is normally distributed.

1. State the claim mathematically. Identify H0 and Ha.
2. Determine when a type I or type II error occurs.
3. Determine whether the hypothesis test is a one-tailed or a two-tailed test and whether to use a z-test or a t-test. Explain your reasoning.
4. If necessary, find the critical value(s) and identify the rejection region(s).
5. Find the appropriate test statistic. If necessary, find the P-value.
6. Decide whether to reject or fail to reject the null hypothesis. Then interpret the decision in the context of the original claim.

11. A tourist agency in Massachusetts claims the mean daily cost of meals and lodging for a family of four traveling in the state is $276. You work for a consumer protection advocate and want to test this claim. In a random sample of 35 families of four traveling in Massachusetts, the mean daily cost of meals and lodging is $285 and the standard deviation is $30. Do you have enough evidence to reject the agency's claim? Use a P-value and a = 0.05.

1. State the claim mathematically. Identify H0 and Ha.
2. Determine when a type I or type II error occurs.
3. Determine whether the hypothesis test is a one-tailed or a two-tailed test and whether to use a z-test or a t-test. Explain your reasoning.
4. If necessary, find the critical value(s) and identify the rejection region(s).
5. Find the appropriate test statistic. If necessary, find the P-value.
6. Decide whether to reject or fail to reject the null hypothesis. Then interpret the decision in the context of the original claim.

Measures of Regression and Prediction Intervals

12. Leisure Hours? Construct a 90% prediction interval for the median number of leisure hours per week when the median number of work hours per week is 45.1.

Median no. of work hrs. per week, x 40.6 43.1 46.9 47.3 46.8
Median no. of leisure hrs. per week, y 26.2 24.3 19.2 18.1 16.6

Median no. of work hrs. per week,x 48.7 50.0 50.7 50.6 50.8
Median no. of leisure hrs. per week,y 18.8 18.8 19.5 19.2 19.5

13. What is the Error of Estimate ? Find the standard error of estimate se and interpret the results.

14. Peanut Yield ? The equation used to predict peanut yield (in pounds) is given (please refer to the attachment for the data). x1 is the number of acres planted (in thousands) and x2 is the number of acres harvested (in thousands). (Source: U.S. National Agricultural Statistics Service)
a. x1 = 1458, x2 = 1450
b. x1 = 1500, x2 = 1475
c. x1 = 1400, x2 = 1385
d. x1 = 1525, x2 = 1500

15. Rice Yield ? To predict the annual rice yield (in pounds), use the equation, where x1 is the number of acres planted (in thousands) and x2 is the number of acres harvested (in thousands). (Source: U.S. National Agricultural Statistics Service)

a. x1 = 2532, x2 = 2255
b. x1 = 3581, x2 = 3021
c. x1 = 3213, x2 = 3065
d. x1 = 2758, x2 = 2714

16. Black Cherry Tree Volume ? The volume (in cubic feet) of black cherry trees can be modeled by the equation, where x1 is the tree's height (in feet) and x2 is the tree's diameter (in inches). (Source: Journal of the Royal Statistical Society)

a. x1 = 70, x2 = 8.6
b. x1 = 65, x2 = 11.0
c. x1 = 83, x2 = 17.6
d. x1 = 87, x2 = 19.6

17. Earnings Per Share ? The earnings per share (in dollars) for McDonald's Corporation are given by the equation where x1 represents total revenue (in billions of dollars) and x2 represents total net worth (in billions of dollars). (Source: McDonald's Corporation)

a. x1 = 11.4,. x2 = 8.6
b. x1 = 8.1, x2 = 6.2
c. x1 = 10.7, x2 = 8.5
d. x1 = 7.3, x2 = 5.1

18. The table lists the personal income and outlays (both in trillions of dollars) for Americans for 11 recent years.
Construct a 95% prediction interval for personal outlays when personal income is 6.4 trillion dollars. Interpret the results.

Personal income, x Personal outlays, y
4.5 3.7
4.9 4.0
5.0 4.1
5.3 4.3
5.6 4.6
5.9 4.8
6.2 5.1
6.5 5.4
7.0 5.7
7.4 6.1
7.8 6.5