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Chapter 7: Q.7.32 (page 300)

You have seen that the larger the sample size, the smaller the sampling error tends to be in estimating a population means by a sample mean. This fact is reflected mathematically by the formula for the standard deviation of the sample mean: 蟽i=蟽/n. For a fixed sample size, explain what this formula implies about the relationship between the population standard deviation and sampling error.

Short Answer

Expert verified

The standard deviation of the sample mean and the standard deviation of the population is thus directly proportional.

Step by step solution

01

Given Information

蟽x炉=蟽/n is the formula for the sample mean's standard deviation.

02

Explanation

The standard deviation of the sample mean determines the degree of sampling error to be expected when a population mean is approximated using a sample mean. The sample size (n)is assumed to be constant. In other words, the denominator in the 蟽x炉formula remains unchanged.

The standard deviation of the sample mean and the standard deviation of the population are thus directly proportional.

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Most popular questions from this chapter

7.48 Menopause in Mexico. In the article "Age at Menopause in Puebla. Mexico" (Human Biology, Vol. 75, No, 2, Pp. 205-206), authors L. Sievert and S. Hautaniemi compared the age of menopause for different populations. Menopause, the last menstrual period, is a universal phenomenon among females. According to the article, the mean age of menopause, surgical or natural, in Puebla, Mexico is 44.8years with a standard deviation of 5.87years. Let x~denote the mean age of menopause for a sample of females in Puebla, Mexico.
a. For samples of size 40, find the mean and standard deviation of x炉. Interpret your results in words.
b. Repeat part (a) with n=120.

A variable of a population has mean 渭 and standard deviation蟽. that For a large sample size n, answer the following questions.

a. Identify the distribution ofx.

b. Does your answer to part (a) depend on n being large? Explain your answer.

c. Identify the mean and the standard deviation ofx.

d. Does your answer to part (c) depend on the sample size being large? Why or why not?

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

7.1 Why is sampling often preferable to conducting a census for the purpose of obtaining information about a population?

What is the sampling distribution of a statistic? Why is it important?
See all solutions

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