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Chapter 7: Q. 7.26 (page 300)

Although, in general, you cannot know the sampling distribution of the sample mean exactly, by what distribution can you often approximate it?

Short Answer

Expert verified

The sampling distribution of the sample mean is often approximated using the normal distribution.

Step by step solution

01

Step 1. Introduction

For a variable x and a given sample size, the distribution of the variable x炉is called the sampling distribution of a sample mean.

It is the distribution of all possible sample means for samples of a given size.

02

Step 2. Explanation

Generally, the value of sampling distribution of a sample mean is not known exactly.

But the value can be often approximated by a normal distribution.

That is under certain conditions, the variable x炉is approximately normally distributed.

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Most popular questions from this chapter

Refer to Exercise 7.9 on page 295.

a. Use your answers from Exercise 7.9(b) to determine the mean, 渭s, of the variable x炉for each of the possible sample sizes.

b. For each of the possible sample sizes, determine the mean, 渭s, of the variable x炉, using only your answer from Exercise 7.9(a).

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 1.40kg and a standard deviation of 0.11kg

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. 7.4on page 304 .

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

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

Why is obtaining the mean and standard deviation of x炉 a first step in approximating the sample distribution of the sample mean by a normal distribution?

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 \(55.4thousand. Assume a standard deviation of \)9.2thousand. 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 64Interpret your answer in terms of the distribution of all possible sample mean salaries for samples of 64classroom teachers.

b. Repeat part (a) for samples of size256

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 64classroom teachers will be at most \(1000?

e. Repeat part (d) for samples of size\)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 \(1,150, determine the probability that the mean tariff rate of 500randomly selected railroad car shipments of ethanol will be within \)100of the mean tariff rate of all railroad-car shipments of ethanol. Interpret your answer in terms of sampling error.
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