# sampling distribution of the mean.

**Instructions:** Complete the following exercises located at the end of each chapter and put them into a Word document to be submitted as directed by the instructor.

Show all relevant work; use the equation editor in Microsoft Word when necessary.

**Chapter 9, numbers 9.7, 9.8, 9.9, 9.13, and 9.14**

*9.7 Define the sampling distribution of the mean.

*9.8 Specify three important properties of the sampling distribution of the mean.

9.9 Indicate whether the following statements are true or false. If we took a random sample of 35 subjects from some population, the associated sampling distribution of the mean would have the following properties:

(a) Shape would approximate a normal curve.

(b) Mean would equal the one sample mean.

(c) Shape would approximate the shape of the population.

(d) Compared to the population variability, the variability would be reduced by a factor equal to the square root of 35.

(e) Mean would equal the population mean.

(f) Variability would equal the population variability.

*9.13 Given a sample size of 36, how large does the population standard deviation have to be for the standard error to be

(a) 1?

(b) 2?

(c) 5?

(d) 100?

*9.14 (a) A random sample of size 144 is taken from the local population of grade-school children. Each child estimates the number of hours per week spent watching TV. At this point, what can be said about the sampling distribution?

(b) Assume that a standard deviation, σ, of 8 hours describes the TV estimates for the local population of schoolchildren. At this point, what can be said about the sampling distribution?

(c) Assume that a mean, µ, of 21 hours describes the TV estimates for the local population of schoolchildren. Now what can be said about the sampling distribution?

(d) Roughly speaking, the sample means in the sampling distribution should deviate, on average, about ___ hours from the mean of the sampling distribution and from the mean of the population.

(e) About 95 percent of the sample means in this sampling distribution should be between ___ hours and ___ hours.

**Chapter 10, numbers 10.9, 10.10, 10.11, and 10.12**

*10.9 The normal range for a widely accepted measure of body size, the body mass index (BMI), ranges from 18.5 to 25. Using the midrange BMI score of 21.75 as the null hypothesized value for the population mean, test this hypothesis at the .01 level of significance given a random sample of 30 weight-watcher participants who show amean BMI = 22.2 and a standard deviation of 3.1.

*10.10 Let’s assume that, over the years, a paper and pencil test of anxiety yields a mean score of 35 for all incoming college freshmen. We wish to determine whether the scores of a random sample of 20 new freshmen, with a mean of 30 and a standard deviation of 10, can be viewed as coming from this population. Test at the .05 level of significance.

*10.11 According to the California Educational Code (http://www.cde.ca.gov/ls/fa/sf/pegui-demidhi.asp), students in grades 7 through 12 should receive 400 minutes of physical education every 10 school days. A random sample of 48 students has a mean of 385 minutes and a standard deviation of 53 minutes. Test the hypothesis at the .05 level of significance that the sampled population satisfies the requiremen

*10.12 According to a 2009 survey based on the United States census (http://www.census.gov/prod/2011pubs/acs-15.pdf), the daily one-way commute time of U.S. workers averages 25 minutes with, we’ll assume, a standard deviation of 13 minutes. An investigator wishes to determine whether the national average describes the mean commute time for all workers in the Chicago area. Commute times are obtained for a random sample of 169 workers from this area, and the mean time is found to be 22.5 minutes. Test the null hypothesis at the .05 level of significance.

Chapter 11, numbers 11.11, 11.19, and 11.20

*11.11 Give two reasons why the research hypothesis is not tested directly.

*11.19 How should a projected hypothesis test be modified if you’re particularly concerned about

(a) the type I error?

(b) the type II error?

*11.20 Consult the power curves in Figure 11.7 to estimate the approximate detection rate, rounded to the nearest tenth, for each of the following situations:

(a) a four-point effect, with a sample size of 13

(b) a ten-point effect, with a sample size of 29 (c) a seven-point effect with a sample size of 18 (Interpolate)

Chapter 12, numbers 12.7, 12.8, and 12.10

*12.7 In Question 10.5 on page 191, it was concluded that, the mean salary among the population of female members of the American Psychological Association is less than that ($82,500) for all comparable members who have a doctorate and teach full time.

(a) Given a population standard deviation of $6,000 and a sample mean salary of $80,100 for a random sample of 100 female members, construct a 99 percent confidence interval for the mean salary for all female members

(b) Given this confidence interval, is there any consistent evidence that the mean salary for all female members fall below $82,500, the mean salary for all members? Answers on page 435.

* 12.8 In Review Question 11.12 on page 218, instead of testing a hypothesis, you might prefer to construct a confidence interval for the mean weight of all 2-pound boxes of candy during a recent production shift.

(a) Given a population standard deviation of .30 ounce and a sample mean weight of 33.09 ounces for a random sample of 36 candy boxes, construct a 95 percent confidence interval.

(b) Interpret this interval, given the manufacturer’s desire to produce boxes of candy that, on the average, exceed 32 ounces.

*12.10 Imagine that one of the following 95 percent confidence intervals estimates the effect of vitamin C on IQ scores:

95 % confidence interval | Lower Limit | Upper Limit |

1 | 100 | 102 |

2 | 95 | 99 |

3 | 102 | 106 |

4 | 90 | 111 |

5 | 91 | 98 |

(a) Which one most strongly supports the conclusion that vitamin C increases IQ scores?

(b) Which one implies the largest sample size?

(c) Which one most strongly supports the conclusion that vitamin C decreases IQ scores?

(d) Which one would most likely stimulate the investigator to conduct an additional experiment using larger sample sizes? Answers on