Question

during a wisdom teeth removal procedure, 1, 2, 3, or 4 wisdom teeth are removed, depending on the patients needs. records indicate that nationwide, the mean number of wisdom teeth removed in a procedure is \\( \mu = 2.25 \\), with a standard deviation of \\( \sigma = 0.73 \\). suppose that we will take a random sample of 8 wisdom teeth removal procedures and record the number of wisdom teeth removed in each procedure. let \\( \overline { x } \\) represent the sample mean of the 8 procedures. 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

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 \(\mu = 2.25\), so \(\mu_{\bar{x}}=2.25\)

Step2: Find the standard deviation of the sampling distribution

The standard deviation of the sampling distribution of the sample mean \(\sigma_{\bar{x}}\) is given by the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)
where \(\sigma = 0.73\) and \(n = 8\)
\(\sigma_{\bar{x}}=\frac{0.73}{\sqrt{8}}\)
\(=\frac{0.73}{2.8284}\)
\(\approx0.26\)

Answer:

(a) \(\mu_{\bar{x}} = 2.25\)
(b) \(\sigma_{\bar{x}}\approx0.26\)