# Deﬁne the sampling distribution of the mean.

**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.**

9.7 Deﬁne the sampling distribution of the mean.

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

9.9 If we took a random sample of 35 subjects from some population, the associated sampling distribution of the mean would have the following properties (true or false).

(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 in order 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, □ , 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, □ , 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.

**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 mid-range BMI score of 21.75 as the null hypothesized value for the population mean, test this hypothesis at the .01 level of signiﬁ cance given a random sam-ple of 30 weight-watcher participants who show a mean BMI 5 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 deter-mine 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/peguidemidhi.asp ), students in grades 7–12 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 requirement.

**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.

**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)

***12.7** In Question 10.5 on page 231, 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 falls below $82,500, the mean salary for all members?

**10.5** NULL HYPOTHESIS ( H 0 ) Once the problem has been described, it must be translated into a statistical hypothesis regarding some population characteristic. Abbreviated as H 0 , the null hypothesis becomes the focal point for the entire test procedure (even though we usually hope to reject it). In the test with SAT scores, the null hypothesis asserts that, with respect to the national average of 500, nothing special is happening to the mean score for the local population of freshmen. An equivalent statement, in symbols, reads: where H 0 represents the null hypothesis and m is the population mean for the local freshman class. Generally speaking, the null hypothesis, H 0 , is a statistical hypothesis that usually asserts that nothing special is happening with respect to some characteristic of the underlying population . Because the hypothesis testing procedure requires that the hypothesized sampling distribution of the mean be centered about a single number (500), the null hypothesis equals a single number ( H 0 : m 5 500). Furthermore, the null hypothesis always makes a precise statement about a characteristic of the population, never about a sample. Remember, the purpose of a hypothesis test is to determine whether a particular outcome, such as an observed sample mean, could have reasonably originated from a population with the hypothesized characteristic. Finding the Single Number for H 0 The single number actually used in H 0 varies from problem to problem. Even for a given problem, this number could originate from any of several sources. For instance, it could be based on available information about some relevant population other than the target population, as in the present example in which 500 reﬂects the mean SAT reading scores for all college-bound students during a recent year. It also could be based on some existing standard or theory—for example, that the mean reading score for the current population of local fresh-men should equal 540 because that happens to be the mean score achieved by all local freshmen during recent years.

**12.8** In Review Question 11.12 on page 263, 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 d e creases IQ scores?

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

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