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Chapter 7: Problem 29

A population has a distribution that is skewed to the right. A sample of size \(n\) is selected from this population. Describe the shape of the sampling distribution of the sample mean for each of the following cases. a. \(n=25\) b. \(n=80\) c. \(n=29\)

Short Answer

Expert verified
For \(n = 25\), the sampling distribution may still show right-skewness due to the skewness in population and possibly insufficient sample size. For \(n = 80\), the Central Limit Theorem will apply and the sampling distribution will look approximately normal, with reduced skewness. The case \(n = 29\) is borderline, and the distribution might look more or less normal, depending on the skewness of the population distribution.

Step by step solution

01

Analyze Sample with Size \(n = 25\)

With a sample size of \(n = 25\), the distribution may not be adequately large to assure the sample means to approximate a normal distribution, especially with the skewed population. Hence, the sample mean distribution might still show some right skewness.
02

Analyze Sample with Size \(n = 80\)

For \(n = 80\), the sample size is sufficient to invoke the Central Limit Theorem. This means, despite skewness in the population, the distribution of the sample mean will look fairly normal, reducing the original skewness.
03

Analyze Sample with Size \(n = 29\)

Sample size \(n = 29\) is a borderline case. For some distributions, this might be enough to consider the sample mean distribution as approximately normal, while for others it might not be quite enough, leaving some residual skewness in the sample mean distribution.

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