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 77 points out of 100, with a standard deviation of 17 points. complete parts (a) through (c) below. choose the correct variable below. a. the variable is the number of high school seniors. b. the variable is the number of high school students. c. the variable is the test length. d. the variable is the test score. b. for samples of size 200, find the mean \\( \mu_{\bar{x}} \\) and standard deviation \\( \sigma_{\bar{x}} \\) of all possible sample mean test scores. \\( \mu_{t}=77 \\) (type an integer or a decimal.) \\( \sigma_{x}=1.2 \\) (round to one decimal place as needed.) c. repeat part (b) for samples of size 400. \\( \mu_{x}=\square \\) (type an integer or a decimal.)

Explanation:

Step1: Mean of sample means

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean. Given the population mean is \(77\), so \(\mu_{\bar{x}} = 77\).

Step2: Standard deviation of sample means (for \(n = 200\))

The formula for the standard deviation of the sampling distribution of the sample mean is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\). Here \(\sigma = 17\) and \(n = 200\).

$$ \sigma_{\bar{x}}=\frac{17}{\sqrt{200}}\approx\frac{17}{14.14}\approx1.2 $$

Step3: Repeat for \(n = 400\)

Using the formula \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\) again, with \(\sigma = 17\) and \(n = 400\)

$$ \sigma_{\bar{x}}=\frac{17}{\sqrt{400}}=\frac{17}{20}=0.85 $$

Answer:

b. \(\mu_{\bar{x}} = 77\), \(\sigma_{\bar{x}}\approx1.2\)
c. \(\mu_{\bar{x}} = 77\), \(\sigma_{\bar{x}}= 0.85\)