Chapter 7.3, Problem 59E

a.

To determine

To explain: The reason of not calculating the probability that sample mean is above 240 seconds.

Answer to Problem 59E

Population is neither normally distributed nor sample size is above 30.

Explanation of Solution

Given:

  $\begin{array}{l}\text{Population mean }\left(\mu \right)=225\\ \text{Population standard deviation}\left(\sigma \right)=60\\ \text{Sample size}\left(n\right)=10\end{array}$

To compute 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 computed here.

b.

To determine

To explain: The reason of computing the probability if the sample of 36 songs is taken out using the central limit theorem. Also, compute the probability that mean play time of song is above 240 seconds.

Answer to Problem 59E

The probability is 0.0668

Explanation of Solution

Calculation:

The central limit theorem states that sampling distribution of sample mean follows the normal distribution if the sample size is at least 30, irrespective 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:

  $\begin{array}{c}P\left(\overline{X}>240\right)=P\left(\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}>\frac{240-\mu }{\frac{\sigma }{\sqrt{n}}}\right)\\ =P\left(Z>\frac{240-225}{\frac{60}{\sqrt{36}}}\right)\\ =P\left(Z>1.50\right)\left(\text{From standard normal table}\right)\\ =0.0668\end{array}$

Thus, the required probability is 0.0668.