Question

the governor of a state wanted to assess the public education system of the state. she instructed a group of researchers to create a test to assess the knowledge of every high school senior in the state. the researchers administered the tests and found that the average score was 88 points out of 100, with a standard deviation of 28 points. complete parts (a) through (c) below. b. the variable is the test length. c. the variable is the test score. d. the variable is the number of high school seniors. b. for samples of size 225, find the mean \\( \mu_{\bar{x}} \\) and standard deviation \\( \sigma_{\bar{x}} \\) of all possible sample mean test scores. \\( \mu_{\bar{x}}=88 \\) (type an integer or a decimal.) \\( \sigma_{\bar{x}}= \\) (round to one decimal place as needed.)

Explanation:

Step1: Recall the formula for the standard deviation of the sample mean

The formula for the standard deviation of the sample mean (\(\sigma_{\bar{x}}\)) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size.

Step2: Substitute the given values into the formula

Given that \(\sigma = 28\) and \(n=225\). First, calculate \(\sqrt{n}=\sqrt{225} = 15\). Then \(\sigma_{\bar{x}}=\frac{28}{15}\approx1.9\)

Answer:

\(\sigma_{\bar{x}}\approx1.9\)