# Chapter 7: The Sampling Distribution of the Sample Mean

7.40

Refer to Exercise $7.10$on page $295$.

a. Use your answers from Exercise $7.10\left(b\right)$to determine the mean, ${\mu}_{i}$, of the variable $\hat{x}$for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, ${\mu}_{i+}$of the variable $\hat{x}$, using only your answer from Exercise $7.10\left(a\right)$

7.41

The winner of the 2012-2013 National Basketball Association (NBA) championship was the Miami Heat. One possible starting lineup for that team is as follows.

a. Determine the population mean height, $\mu $, of the five players:

b. Consider samples of size $2$without replacement. Use your answer to Exercise $7.11\left(b\right)$on page $295$and Definition $3.11$on page $140$to find the mean, ${\mu}_{\mathrm{r}}$, of the variable $\hat{x}$.

c. Find ${\mu}_{{x}^{*}}$using only the result of part *(a)*.

Q 10RP.

Hours Actually Worked. In the article "How Hours of Work Affect Occupational Earnings" (Monthly Labor Reriew, Vol. 121), D. Hecker discussed the number of hours actually worked as opposed to the number of hours paid for. The study examines both full-time men and full-time women in $87$ different occupations. According to the article. the mean number of hours (actually) worked by female marketing and advertising managers is $\mu =45$ hours. Assuming a standard deviation of $\sigma =7$ hours, decide whether cach of the following statements is true or false or whether the information is insufficient to decide. Give a reason for each of your answers.

a. For a random sample of $196$ female marketing and advertising managers, chances are roughly $95\%$ that the sample mean number of hours worked will be between $31$ hours and $59$ hours.

b. Approximately $95\%$ of all possible observations of the number of hours worked by female marketing and advertising managers lie between 31 hours and $59$ hours.

c. For a random sample of $196$ female marketing and advertising managers, chances are roughly $95\%$ that the sample mean number of hours worked will be between $44$ hours and $46$ hours.

Q 11RP.

Hours Actually Worked. Repeat Problem $10$, assuming that the number of hours worked by female marketing and advertising managers is normally distributed.

Q 12RP.

Western Pygmy-Possum. The foraging behavior of the western pygmy-possum was investigated in the article "Strategies of a Small Nectarivorous Marsupial, the Western Pygmy-Possum, in Response to Seasonal Variation in Food Availability" (Journal of Mammalogy, Vol. 96, No. 6, pp. 1525-1535) by D. Morrant and S. Petit. The weights of adult male pygmy-possums in Australia are normally distributed with a mean of $8.5\mathrm{g}$ and a standard deviation of $0.3\mathrm{g}$

a. Sketch the normal curve for the pygmy-possum weights.

b. Find the sampling distribution of the sample mean for samples of size $4$Draw a graph of the normal curve associated with $\overline{x}$

c. Repeat part (b) for samples of size $9$

Q 13RP.

Western Pygmy-Possum. Refer to Problem $12$

a. Find the percentage of all samples of four pygmy possums that have mean weights within $0.225\mathrm{g}$the population mean weight of $8.5\mathrm{g}$

b. Obtain the probability that the mean weight of four randomly selected pygmy possums will be within $0.225\mathrm{g}$the population mean weight of $8.5\mathrm{g}$

c. Interpret the probability you obtained in part (b) in terms of sampling error.

d. Repeat parts (a) -(c) for samples of size $9$

Q 14RP.

Blood Glucose Level. In the article "Drinking Glucose Improves Listening Span in Students Who Miss Breakfast" (Educational Research, Vol. 43, No. 2, pp. $201-207$). authors N. Morris and P. Sarll explored the relationship between students who skip breakfast and their performance on a number of cognitive tasks. According to their findings, blood glucose levels in the morning, after a $9-$hour fast, have a mean of $4.60\mathrm{mmol}/\mathrm{L}$with a standard deviation of $0.16\mathrm{mmol}/\mathrm{L}$. (Note; $mmol/L$is an abbreviation of millimoles/liter. which is the world standard unit for measuring glucose in the blood.)

a. Determine the sampling distribution of the sample mean for samples of size $60$

b. Repeat part (a) for samples of size $120$

c. Must you assume that the blood glucose levels are normally distributed to answer parts (a) and (b)? Explain your answer.

Q. 15

The American Council of Life Insurers provides information about life insurance in force per covered family in the Life Insurers Fact Book. Assume that standard deviation of life insurance in force is \($50,900\).

a. Determine the probability that the sampling error made in estimating the population mean life insurance in force by that of a sample of \(500\) covered families will be \($2000\) or less.

b. Must you assume that life-insurance amounts are normally distributed in order to answer part (a)? What if the sample size is \(20\) instead of \(500\)?

c. Repeat part (a) for a sample size of \(5000\).

Q 16RP.

Paint Durability. A paint manufacturer in Pittsburgh claims that his paint will last an average of $5$ years. Assuming that paint life is normally distributed and has a standard deviation of $0.5$ year. answer the following questions:

a. Suppose that you paint one house with the paint and that the paint lasts $4.5$ years. Would you consider that evidence against the manufacturer's claim? (Hint: Assuming that the manufacturer's claim is correct, determine the probability that the paint life for a randomly selected house painted with the paint is $4.5$ years or less.)

b. Suppose that you paint $10$ houses with the paint and that the paint lasts an average of $4.5$ years for the $10$ houses. Would you consider that evidence against the manufacturer's claim?

c. Repeat part (b) if the paint lasts an average of $4.9$ years for the $10$ houses painted.

Q 17RP.

Cloudiness in Breslau. In the paper "Cloudiness: Note on a Novel Case of Frequency" (Proceedirgs of the Royal Society of London, Vol. 62. pp. 287-290), K. Pearson examined data on daily degree of cloudiness, on a scale of $0$ to $10$ , at Breslau (Wroclaw), Poland, during the decade $1876-1885$. A frequency distribution of the data is presented in the following table. From the table, we find that the mean degree of cloudiness is $6.83$ with a standard deviation of $4.28$.

a. Consider simple random samples of $100$ days during the decade in question. Approximately what percentage of such samples have a mean degree of cloudiness exceeding $7.5$ ?

b. Would it be reasonable to use a normal distribution to obtain the percentage required in part (a) for samples of size $5$ ? Explain your answer.