91Ó°ÊÓ

7.67 Brain Weights. In 1905, R. Pearl published the article "Biometrical Studies on Man. 1. Variation and Correlation in Brain Weight" (Biometrika, Vol. 4, pp. 13-104). According to the study, brain weights of $S$ wedish men are normally distributed with a mean of $1.40\mathrm{kg}$ and a standard deviation of $0.11\mathrm{kg}$

a. Determine the sampling distribution of the sample mean for samples of size $3$ Interpret your answer in terms of the distribution of all possible sample mean brain weights for samples of three Swedish men.

b. Repeat part (a) for samples of size $12$

c. Construct graphs similar to those shown in Fig. $7.4$on page 304 .

d. Determine the percentage of all samples of three Swedish men that have mean brain weights within $0.1\mathrm{kg}$ of the population mean brain weight of $1.40\mathrm{kg}$. Interpret your answer in terms of sampling error.

e. Repeat part (d) for samples of size $12$

Teacher Salaries. Data on salaries in the public school system are published annually in Ranking of the States and Estimates of School Statistics by the National Education Association. The mean annual salary of (public) classroom teachers is $\backslash (55.4$thousand. Assume a standard deviation of $\backslash )9.2$thousand. Do the following tasks for the variable "annual salary" of classroom teachers.

a. Determine the sampling distribution of the sample mean for samples of size $64$Interpret your answer in terms of the distribution of all possible sample mean salaries for samples of $64$classroom teachers.

b. Repeat part (a) for samples of size$256$

c. Do you need to assume that classroom teacher salaries are normally distributed to answer parts (a) and (b)? Explain your answer.

d. What is the probability that the sampling error made in estimating the population means salary of all classroom teachers by the mean salary of a sample of $64$classroom teachers will be at most $\backslash (1000$?

e. Repeat part (d) for samples of size$\backslash )256$

Ethanol Railroad Tariffs. An ethanol railroad tariff is a fee charged for shipments of ethanol on public railroads. The Agricultural Marketing Service publishes tariff rates for railroad-car shipments of ethanol in the Biofuel Transportation Database. Assuming that the standard deviation of such tariff rates is $\backslash (1,150$, determine the probability that the mean tariff rate of $500$randomly selected railroad car shipments of ethanol will be within $\backslash )100$of the mean tariff rate of all railroad-car shipments of ethanol. Interpret your answer in terms of sampling error.

The following graph shows the curve for a normally distributed variable. Superimposed are the curves for the sampling distributions of the sample mean for two different sample sizes.

a. Explain why all three curves are centered at the same place.

b. Which curve corresponds to the larger sample size? Explain your answer.

c. Why is the spread of each curve different?

d. Which of the two sampling-distribution curves corresponds to the sample size that will tend to produce less sampling error? Explain your answer.

c. Why are the two sampling-distribution curves normal curves?