Question

the state of california has studied a large population of pregnant women applying for maternity leave. the women stated which trimester of their pregnancy (1, 2, or 3) they are in. the mean response was \\( \mu = 1.70 \\), with a standard deviation of \\( \sigma = 0.77 \\). suppose that we will take a random sample of \\( n = 10 \\) of these womens responses. let \\( \overline { x } \\) represent the sample mean of the 10 womens responses. 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 of the sample mean

The mean of the sampling distribution of the sample mean \(\mu_{\bar{x}}\) is equal to the population mean \(\mu\).
Given \(\mu = 1.70\), so \(\mu_{\bar{x}}=\mu = 1.70\)

Step2: Find the standard deviation of the sampling distribution of the sample mean

The formula for the standard deviation of the sampling distribution of the sample mean (also known as the standard error) is \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\)
Given \(\sigma = 0.77\) and \(n = 10\)

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Answer:

(a) \(\mu_{\bar{x}} = 1.70\)
(b) \(\sigma_{\bar{x}}\approx0.24\)