Question

(a) what is the value of \\( \mu _ { x } \\)?
the value of \\( \mu _ { x } \\) is 400
(b) what is the value of \\( \sigma _ { x } \\)?
the value of \\( \sigma _ { x } \\) is 30
(c) if the sample size is \\( n = 9 \\), what is likely true about the shape of the population?
a. the shape of the population is skewed left.
b. the shape of the population is skewed right.
c. the shape of the population is approximately normal
d. the shape of the population cannot be determined
(d) if the sample size is \\( n = 9 \\), what is the standard deviation of the population from which the sample was drawn?
the standard deviation of the population from which the sample was drawn is

Explanation:

Step1: Recall the formula for the standard deviation of the sampling distribution

The formula for the standard deviation of the sampling distribution 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: Rearrange the formula to solve for \(\sigma\)

From \(\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}\), we can solve for \(\sigma\) by multiplying both sides of the equation by \(\sqrt{n}\). So, \(\sigma = \sigma_{\bar{x}}\times\sqrt{n}\).

Step3: Substitute the known values

We know that \(\sigma_{\bar{x}} = 30\) (from part (b)) and \(n = 9\). Substituting these values into the formula \(\sigma=\sigma_{\bar{x}}\times\sqrt{n}\), we get \(\sigma=30\times\sqrt{9}\).

Step4: Calculate the value of \(\sigma\)

Since \(\sqrt{9}=3\), then \(\sigma = 30\times3=90\).

Answer:

90