the state of colorado has studied a large population of pregnant women applying for maternity leave. the…

the state of colorado 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 $mu = 2.67$, with a standard deviation of $sigma = 0.79$. suppose that we will take a random sample of $n = 6$ of these womens responses. let $overline{x}$ represent the sample mean of the 6 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.\n(a) find $mu_{overline{x}}$ (the mean of the sampling distribution of the sample mean).\n$mu_{overline{x}}=$\n(b) find $sigma_{overline{x}}$ (the standard deviation of the sampling distribution of the sample mean).\n$sigma_{overline{x}}=$

the state of colorado 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 $mu = 2.67$, with a standard deviation of $sigma = 0.79$. suppose that we will take a random sample of $n = 6$ of these womens responses. let $overline{x}$ represent the sample mean of the 6 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.\n(a) find $mu_{overline{x}}$ (the mean of the sampling distribution of the sample mean).\n$mu_{overline{x}}=$\n(b) find $sigma_{overline{x}}$ (the standard deviation of the sampling distribution of the sample mean).\n$sigma_{overline{x}}=$

Answer

Explanation:

Step1: Recall the mean of sampling - distribution property

The mean of the sampling distribution of the sample mean $\mu_{\bar{x}}$ is equal to the population mean $\mu$. Given $\mu = 2.67$, so $\mu_{\bar{x}}=\mu$. $\mu_{\bar{x}}=2.67$

Step2: Recall the standard - deviation of sampling - distribution formula

The standard deviation of the sampling distribution of the sample mean (also known as the standard error) is given by the formula $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Here, $\sigma = 0.79$ and $n = 6$. $\sigma_{\bar{x}}=\frac{0.79}{\sqrt{6}}$ $\sigma_{\bar{x}}=\frac{0.79}{2.44949}\approx0.32$

Answer:

(a) $\mu_{\bar{x}} = 2.67$ (b) $\sigma_{\bar{x}} = 0.32$