Chapter 7.3, Problem 55E

To determine

To find: the number of song need to sample if the standard deviation of the sampling distribution of $\overline{x}$ to be 10 seconds.

Answer to Problem 55E

36 songs

Explanation of Solution

Given:

  $\begin{array}{c}\text{μ=225}\text{seconds}\\ \text{σ}\text{= 60}\text{seconds}\\ {\text{σ}}_{\text{<mover accent="true"> <mo>x</mo> <mo>¯</mo></mover>}}\text{=}\text{10}\text{seconds}\end{array}$

Formula used:

Calculation:

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

  ${\text{σ}}_{\text{<mover accent="true"> <mo>x</mo> <mo>¯</mo></mover>}}=\frac{\sigma }{\sqrt{n}}$

Multiplying both the side by $\sqrt{n}$

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

Dividing both the side by ${\text{σ}}_{\text{<mover accent="true"><mo> x</mo> <mo>¯</mo></mover>}}$

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

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 songs if the standard deviation of the sampling distribution of the sample means to be 10 seconds