Q.51 - The Practice of Statistics for AP

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.

Step-by-Step Solution

Answer

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

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 $60 for \sigma and 30 for {\sigma }_{\overline{x}}$ in the aboveexpression and simplify for $n$

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

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