Question

a large airline company called tale winds monitors customer satisfaction by asking customers to rate their experience as a 1, 2, 3, 4, or 5, where a rating of 1 means \very poor\ and 5 means \very good\. the customers ratings have a population mean of \\( \mu = 4.14 \\), with a population standard deviation of \\( \sigma = 1.41 \\). suppose that we will take a random sample of \\( n = 7 \\) customers ratings. let \\( \overline { x } \\) represent the sample mean of the 7 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: Find the mean of the sampling distribution of the sample mean

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\).
\(\mu_{\bar{x}}=\mu\)
Given \(\mu = 4.14\), so \(\mu_{\bar{x}}=4.14\)

Step2: Find the standard deviation of the sampling distribution of the sample mean

The formula for the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)
Given \(\sigma = 1.41\) and \(n = 7\)
\(\sigma_{\bar{x}}=\frac{1.41}{\sqrt{7}}\)
First, calculate \(\sqrt{7}\approx2.6458\)
Then \(\sigma_{\bar{x}}=\frac{1.41}{2.6458}\approx0.53\)

Answer:

(a) \(\mu_{\bar{x}} = 4.14\)
(b) \(\sigma_{\bar{x}}\approx0.53\)