Question

Songs on an iPod Refer to Exercise 53 . How many songs would you need to sample if you wanted the standard deviation of the sampling distribution of${\mathrm{x}}^{-\overline{x}}$ to be 10 seconds? Justify your answer.

Short answer

The sample should be 36 tracks if the sampling distribution's standard deviation is 10 seconds.

Step-by-step solution

Given information 

Given,

$$\begin{gathered} \mu =225\text{seconds} \\ \sigma =60\text{seconds} \\ {\sigma }_{=}=10\text{seconds} \end{gathered}$$

Calculation

The standard deviation of the sampling distribution of the sample mean is the population standard deviation divided by the square root of the sample. $\overline{x}$

${\sigma }_1=\frac{\sigma }{\sqrt{n}}$

Now Multiply on both the side by $\sqrt{n}$

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

Now Divide on both the side by ${\sigma }_2$

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

Squaring both the sides

$n=\frac{{\sigma }^2}{{\sigma }_{\overline{x}}^2}$

Calculating the value of $n$

$n=\frac{{\sigma }^2}{{\sigma }_{\overline{x}}^2}=\frac{{60}^2}{{10}^2}=\frac{3600}{100}=36$

So the sample should be 36 tracks if the sampling distribution's standard deviation is 10 seconds.