Chapter 7.3, Problem 53E

(a)

To determine

To find: the mean of the sampling distribution of $\overline{x}$

Answer to Problem 53E

225 seconds

Explanation of Solution

Given:

  $\begin{array}{l}\mu =225\\ \sigma =60\\ n=10\end{array}$

Calculation:

The mean of the sampling distribution of the sample mean is equal to the population mean:

  ${\mu }_{\overline{x}}=\mu =225$

The units of the mean of the sampling distribution of the sample mean are the sample as the units of the population mean and thus the mean is $225$ second.

(b)

To determine

To Calculate: and interpret the standard deviation of the sampling distribution of $\overline{x}$ and verify the 10% condition is met.

Answer to Problem 53E

18.9737

Explanation of Solution

Given:

  $\begin{array}{l}\mu =225\\ \sigma =60\\ n=10\end{array}$

Formula used:

  ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}$

Calculation:

The mean of the sampling distribution of the sample mean is equal to the population mean:

  ${\mu }_{\overline{x}}=\mu =225$

The standard deviation of the sampling distribution is

  ${\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}=\frac{60}{\sqrt{10}}=18.9737$

The units of the standard deviation of the sampling distribution of the sample mean are the same as the units of the population standard deviation and therefore the standard deviation is $18.9737$ seconds.

The sample mean play time of 10 randomly selected songs of the iPod varies on average by $18.9737$ second from the mean play time $225$ seconds.