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Chapter 7: Q 7.66. (page 307)

According to the central limit theorem, for a relatively large sample size, the variable x~is approximately normally distributed.

a. What rule of thumb is used for deciding whether the sample size is relatively large?

b. Roughly speaking, what property of the distribution of the variable under consideration determines how large the sample size must be for a normal distribution to provide an adequate approximation to the distribution of x~ ?

Short Answer

Expert verified

Part (a)the sample size is greater than30.

Part (b) For a normal distribution to provide an adequate approximation to the distribution of x¯the sample size must be large.

Step by step solution

01

Part (a) Step 1: Given information 

the variablex~is approximately normally distributed.

02

Part (a) Step 2: Concept

Formula used:population mean and standard deviation:μx¯=μandσx¯=σ/n.

03

Part (a) Step 3: Explanation

We consider a sample as a large sample if the sample size is greater than30.

04

Part (b) Step 1: Explanation

The probability density function's symmetry and bell-shapedness determine how large sample size needed to be for a normal distribution to provide a good approximation to the distribution of x¯

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Most popular questions from this chapter

What is another name for the standard deviation of the variable x ? What is the reason for that name?

Population data: 2,5,8

Part (a): Find the mean, μof 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.

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a. Obtain the probability that the weight of one randomly selected bag of water-softener salt will be 39lb or less, if the company's claim is true.

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Explain why increasing the sample size tends to result in a smaller sampling error when a sample means is used to estimate a population mean.
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