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 = 2.63 \\), with a standard deviation of \\( \sigma = 0.53 \\). 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: Recall the formula for 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 \(\mu\).
\(\mu_{\bar{x}}=\mu\)

Step2: Substitute the given population mean

Given \(\mu = 2.63\), so \(\mu_{\bar{x}}=2.63\)

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

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

Step4: Substitute the given values

Given \(\sigma = 0.53\) and \(n = 8\)
\(\sigma_{\bar{x}}=\frac{0.53}{\sqrt{8}}\)
\(\sigma_{\bar{x}}=\frac{0.53}{2.8284}\approx0.19\)

Answer:

(a) \(\mu_{\bar{x}} = 2.63\)
(b) \(\sigma_{\bar{x}}\approx0.19\)