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Chapter 7: Problem 22

A population has a normal distribution. 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. \(n=94\) b. \(n=11\)

Short Answer

Expert verified
For \(n=94\), the sampling distribution of the sample mean will be approximately normal due to the Central Limit Theorem. For \(n=11\), the shape might not be normal because it's not a large enough sample size to assure normality via the Central Limit Theorem, unless the population itself is normally distributed.

Step by step solution

01

Shape of the Sampling Distribution for case a

Given that \(n=94\) which is larger than 30, the sampling distribution of the mean will be approximately normal. The Central Limit theorem assures this.
02

Shape of the Sampling Distribution for case b

In the case of \(n=11\), which is less than 30, we cannot ensure the normality of the sampling distribution of the mean. Although Central Limit theorem works for smaller samples as well, as a rule of thumb, it is commonly used when \(n>30\). If the population from which this sample of 11 observations is drawn is not normal, then the shape can differ significantly from a normal distribution.

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