Question

crave creamery is an ice cream shop chain that has locations all across the nation. customers at crave creamery have the option of ordering 1, 2, or 3 scoops of ice cream in their cone. the mean number of scoops ordered is \\( \mu = 1.50 \\), with a standard deviation of \\( \sigma = 0.72 \\). suppose that we will take a random sample of \\( n = 10 \\) ice cream cone orders and record the number of scoops for each. let \\( \overline { x } \\) represent the sample mean of the number of scoops for the 10 ice cream cone orders. 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 \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\).
\(\mu_{\bar{x}}=\mu\)
Given \(\mu = 1.50\), so \(\mu_{\bar{x}}=1.50\)

Step2: 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 given by \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)
Given \(\sigma = 0.72\) and \(n = 10\)
\(\sigma_{\bar{x}}=\frac{0.72}{\sqrt{10}}\)
\(\sqrt{10}\approx3.1623\)
\(\sigma_{\bar{x}}=\frac{0.72}{3.1623}\approx0.23\)

Answer:

(a) \(\mu_{\bar{x}} = 1.50\)
(b) \(\sigma_{\bar{x}}\approx0.23\)