Question

Songs on an iPod David's iPod has about $10000$songs. The distribution of the play times for these songs is heavily skewed to the right with a mean of $225$seconds and a standard deviation of $60$seconds. How many songs would you need to sample if you wanted the standard deviation of the sampling distribution of $\overline{x}$to be$30$seconds? justiify your answer.

Short answer

The required number of songs to sample is$4$.

Step-by-step solution

Step-1 Given Information

Given in the question that

Number of songs in the iPod $=10000$

Mean$=225$seconds

Standard deviation $=60$seconds

Number of songs in SRS$=10$

we have to find ow many songs would you need to sample if you wanted the standard deviation of the sampling distribution of $\overline{x}$to be$30$seconds.

Step-2 Explanation

The mean of the sampling distribution of $\overbrace{p}is{\mu }_{\overbrace{p}}=p$

The standard deviation of a sample distribution of the sample mean $\overline{x}$if the population standard deviation is $\sigma$can be written as,

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

Here, number of songs to sample is the sample size.

Also,

Population standard deviation, $\sigma =60$

Standard deviation of the sample, ${\sigma }_{\overline{x}}=30$

It is known that the standard deviation of a sample distribution of the sample mean $\overline{x}$if the population standard deviation is $\sigma$can be written as,

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

To find the sample size, use the above formula.

Write the formula as:

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

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

square both sides of the equation

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

substitute $60for\sigma and30for{\sigma }_{\overline{x}}$in the aboveexpression and simplify for $n$

$n=\frac{{60}^2}{{30}^2}=4$

Hence, the requirred number of songs to sample is $4$