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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 a sample is frequently approximated by the Normal Distribution.

Step by step solution

01

Given Information

We have to find the method to approximate the sample size.

02

Explanation

The variable xis nearly normally distributed, according to the central limit theorem, for a relatively large sample size.

The sampling distribution of a sample is frequently approximated by the Normal Distribution.

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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

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

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.

A variable of a population is normally distributed with mean 渭and standard deviation 蟽. For samples of size n, fill in the blanks. Justify your answers.

a. Approximately 68%of all possible samples have means that lie within of the population mean, 渭

b. Approximately 95%of all possible samples have means that lie within of the population mean, 渭

c. Approximately 99.7%of all possible samples have means that lie within of the population mean, 渭

d. 100(1-伪)%of all possible samples have means that lie within _of the population mean, 渭(Hint: Draw a graph for the distribution of x, and determine the z-scores dividing the area under the normal curve into a middle 1-伪area and two outside areas of伪/2

Refer to Exercise 7.6 on page 295.

a. Use your answers from Exercise 7.6(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 7.6(a).
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