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Chapter 7: Q. 7.27 (page 300)

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?

Short Answer

Expert verified

The normal distribution always depends upon two parameters the mean and the standard deviation for any given random variable. Hence, the mean and standard deviation of x炉are obtained first when approximating the sample distribution of the sample mean by a normal distribution.

Step by step solution

01

Step 1. Given Information

The objective is to find the reason why obtaining the mean and standard deviation of x炉is a first step in approximating the sampling distribution of the sample mean by a normal distribution.

02

Step 2. Explanation

A variable is normally distributed if its distribution has the shape of a normal curve and that a normal distribution is determined by the mean and standard deviation.

Hence a first step in learning how to approximate the sampling distribution of the sample mean by a normal distribution is to obtain the mean and standard deviation of the sample mean, that is, of the variable x炉.

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Most popular questions from this chapter

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:

Part (a): Find the population mean height of the five players.

Part (b): For samples of size 2, construct a table similar to Table 7.2 on page 293. Use the letter in parentheses after each player's name to represent each player.

Part (c): Draw a dotplot for the sampling distribution of the sample mean for samples of size 2.

Part (d): For a random sample of size2, what is the chance that the sample mean will equal the population mean?

Part (e): For a random sample of size 2, obtain the probability that the sampling error made in estimating the population mean by the sample mean will be1 inch or less; that is, determine the probability that x will be within1 inch of 渭. Interpret your result in terms of percentages.

A statistic is said to be an unbiased estimator of a parameter if the mean of all its possible values equals the parameter; otherwise, it is said to be a biased estimator. An unbiased estimator yields, on average, the correct value of the parameter, whereas a biased estimator does not.

Part (a): Is the sample mean an unbiased estimator of the population mean? Explain your answer.

Part (b): Is the sample median an unbiased estimator of the population mean? Explain your answer.

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).

A variable of a population has a mean of 渭=35and a standard deviation of 蟽=42.

a. If the variable is normally distributed, identify the sampling distribution of the sample mean for samples of size 9.

b. Can you answer part (a) if the distribution of the variable under consideration is unknown? Explain your answer.

c. Can you answer part (a) if the distribution of the variable under consideration is unknown but the sample size is 36instead of 9?

Why or why not?

In Exercises 7.3-7.10, we have given population data for a variable. For each exercise, do the following tasks.
a. Find the mean, 渭, of the variable.
b. For each of the possible sample sizes, construct a table similar to Table 7.2 on page 293 and draw a dotplot for the sampling distribution of the sample mean similar to Fig. 7.1 on page 293.
c. Construct a graph similar to Fig. 7.3 and interpret your results.
d. For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.
e. For each of the possible sample sizes, find the probability that the sampling error made in estimating the population mean by the sample mean will be 0.5or less (in magnitude), that is, that the absolute value of the difference between the sample mean and the population mean is at most 0.5.
7.4 Population data: 2,5,8.
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