Q.59 - 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. 

(a) Explain why you cannot safely calculate the probability that the mean play time $\overline{x}$ is more than $4$ minutes $\left(240\right)$ seconds for an SRS of $10$ songs.

(b) Suppose we take an SRS of  $36$songs instead. Explain how the central limit theorem allows us to find the probability that the mean playtime is more than $240$ seconds. Then calculate this probability. Show your work.

Step-by-Step Solution

Answer

(a) Population is neither normally distributed nor sample size is above $30$

(b) The probability is $0.0668$

Part (a) Step-1 Given Information

Given in the question that,

Population mean$\mu$$=225$

Population standard deviation$\sigma$$=60$

Sample size$\left(n\right)=10$

we have to find the probability that the mean play time $\overline{x}$ is more than $4$ minutes $\left(240\right)$ seconds for an SRS of $10$ songs.

Part (a) Step-2 Explanation

To calculate the probability either the sample size must be above $30$ or the population must be normally distributed. Here, neither the original population is normally distributed nor the sample size is above $30$Thus, the required probability cannot be  calculated here.

Part (b) Step-1 Given Information

Given in the question that we take an SRS of  $36$songs instead we have to Explain how the central limit theorem allows us to find the probability that the mean playtime is more than $240$ seconds and calculate probability.

Part (b) Step-2 Explanation

The central limit theorem states that sampling distribution of sample mean follows the normal distribution if the sample size is at least $30$ , without considering of the nature of the distribution.

The sample size is higher than $30$ . Thus, the required probability can be calculated here.

The probability that mean play time of song is above $240$ seconds is calculated as follows:

$P\left(\overline{X}>240\right)=P\left(\frac{{\displaystyle \frac{x-\mu }{\sigma }}}{\sqrt{n}}>\frac{{\displaystyle \frac{240-\mu }{\sigma }}}{\sqrt{n}}\right)$

                  $=P\left(Z>\frac{{\displaystyle \frac{240-255}{60}}}{\sqrt{36}}\right)$  $P\left(Z>\frac{{\displaystyle \frac{240-225}{60}}}{\sqrt{36}}\right)$

                 $=P(Z>1.50) \left(From s\tan dard normal table\right)$$=P\left(Z>1.50\right)\left(From s\tan dard normal table\right)$

                  $=0.0668$

Thus, the required probability is $0.0668$