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.

c. the value is the number of high school seniors.

b. for samples of size 225, find the mean \\( \mu _ { \overline { x } } \\) and standard deviation \\( \sigma _ { \overline { x } } \\) of all possible sample mean test scores.

\\( \mu _ { \overline { x } } = 88 \\) (type an integer or a decimal.)

\\( \sigma _ { \overline { x } } = 1.9 \\) (round to one decimal place as needed.)

c. repeat part (b) for samples of size 450.

\\( \mu _ { \overline { x } } = \square \\) (type an integer or a decimal.)

Explanation:

Step1: Recall the formula for the mean of the sample means

The mean of the sample means \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\). Given that the population mean \(\mu = 88\), for any sample size \(n\), \(\mu_{\bar{x}}=\mu\).

Step2: Recall the formula for the standard deviation of the sample means

The formula for the standard deviation of the sample means (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), where \(\sigma\) is the population standard deviation and \(n\) is the sample size. Here, \(\sigma = 28\) and \(n = 450\).

Step3: Calculate \(\mu_{\bar{x}}\)

Since \(\mu_{\bar{x}}=\mu\), and \(\mu = 88\), so \(\mu_{\bar{x}}=88\)

Step4: Calculate \(\sigma_{\bar{x}}\)

Substitute \(\sigma = 28\) and \(n = 450\) into the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\).

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Answer:

\(\mu_{\bar{x}} = 88\)