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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: σi=σ/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

σx¯=σ/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 σx¯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

Loan Amounts. B. Ciochetti et al. studied mortgage loans in the article "A Proportional Hazards Model of Commercial Mortgage Default with Originator Bias" (Joumal of Real Exfate and Economics, Vol, 27. No. 1. pp. 5-23). According to the article, the loan amounts of loans originated by a large insurance-company lender have a mean of \(6.74 million with a standard deviation of \)15.37 million. The variable "loan amount" is known to have a right-skewed distribution.

a. Using units of millions of dollars, determine the sampling distribution of the sample mean for samples of size 200 . Interpret your result.

b. Repeat part (a) for samples of size 600

c. Why can you still answer parts (a) and (b) when the distribution of loan amounts is not normal, but rather right skewed?

d. What is the probability that the sampling error made in estimating the population mean loan amount by the mean loan amount of a simple random sample of 200 loans will be at most $1 million?

e. Repeat part (d) for samples of size 600

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.

Does the sample size have an effect on the mean of all possible sample mean? Explain your answer.

Population data: 3,4,7,8

Part (a): Find the mean, μ, of the variable.

Part (b): For each of the possible sample sizes, construct a table similar to Table 7.2on the page 293and draw a dotplot for the sampling for the sampling distribution of the sample mean similar to Fig 7.1on page 293.

Part (c): Construct a graph similar to Fig 7.3and interpret your results.

Part (d): For each of the possible sample sizes, find the probability that the sample mean will equal the population mean.

Part (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, that is, that the absolute value of the difference between the sample mean and the population mean is at most 0.5.

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 44.8 years with a standard deviation of 5.87 years. Letx¯ 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¯. Interpret your results in words.
b. Repeat part (a) with n=120.
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