Question

easy eats diner is a large restaurant chain. after paying for a meal at easy eats diner, customers are asked to rate the quality of the food as a 1, 2, 3, 4, or 5, where a rating of 1 means
ot good\ and 5 means \excellent\. the customers ratings have a population mean of \\( \mu = 4.50 \\), with a standard deviation of \\( \sigma = 1.26 \\). suppose that we will take a random sample of \\( n = 6 \\) customers ratings. let \\( \overline{x} \\) represent the sample mean of the 6 customers ratings. consider the sampling distribution of the sample mean \\( \overline{x} \\).
complete the following. do not round any intermediate computations. write your answers with two decimal places, rounding if needed.
(a) find \\( \mu_{\overline{x}} \\) (the mean of the sampling distribution of the sample mean).
\\( \mu_{\overline{x}}= \\)
(b) find \\( \sigma_{\overline{x}} \\) (the standard deviation of the sampling distribution of the sample mean).
\\( \sigma_{\overline{x}}= \\)

Explanation:

Step1: Recall the formula for the mean of the sampling distribution of the sample mean

The mean of the sampling distribution of the sample mean \(\mu_{\overline{x}}\) is equal to the population mean \(\mu\).
\(\mu_{\overline{x}}=\mu\)

Step2: Substitute the given population mean

Given \(\mu = 4.50\), so \(\mu_{\overline{x}}=4.50\)

Step3: Recall the formula for the standard deviation of the sampling distribution of the sample mean

The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}\)

Step4: Substitute the given values of \(\sigma\) and \(n\)

Given \(\sigma = 1.26\) and \(n = 6\), then \(\sigma_{\overline{x}}=\frac{1.26}{\sqrt{6}}\)

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Answer:

(a) \(\mu_{\overline{x}} = 4.50\)
(b) \(\sigma_{\overline{x}}\approx0.51\)