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Chapter 7: Problem 33

Among college students who hold part-time jobs during the school year, the distribution of the time spent working per week is approximately normally distributed with a mean of \(20.20\) hours and a standard deviation of \(2.60\) hours. Let \(\bar{x}\) be the average time spent working per week for 18 randomly selected college students who hold part-time jobs during the school year. Calculate the mean and the standard deviation of the sampling distribution of \(\bar{x}\), and describe the shape of this sampling distribution.

Short Answer

Expert verified
The mean (\(\mu_{\bar{x}}\)) of the sample distribution is 20.20 hours, the standard deviation (\(\sigma_{\bar{x}}\)) is approximately 0.61 hours. The shape of this sampling distribution is approximately normal.

Step by step solution

01

Calculate the Mean

The mean of the sampling distribution of the sample mean \(\bar{x}\) is equal to the population mean. Given the population mean is \(20.20\) hours, hence mean of \(\bar{x}\), denoted as \(\mu_{\bar{x}}\), is also \(20.20\) hours. That is, \(\mu_{\bar{x}} = 20.20\).
02

Calculate the Standard Deviation

The standard deviation of the sampling distribution, denoted as \(\sigma_{\bar{x}}\), is the standard deviation of the population divided by the square root of the sample size. Given the population standard deviation is \(2.60\) hours and the sample size is 18, hence \(\sigma_{\bar{x}} \) can be calculated as follows: \(\sigma_{\bar{x}} = \sigma/ \sqrt{n} = 2.60 / \sqrt{18} \approx 0.61\) hours.
03

Shape of the Sampling Distribution

As stated, the distribution of the time spent working per week is approximately normally distributed, and the sample size is reasonably large (n = 18), hence we can say that the sampling distribution of \(\bar{x}\) will also be approximately normally distributed due to the Central Limit Theorem.

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