91影视

Given in the question that,

Population mean$渭$$=225$

Population standard deviation$蟽$$=60$

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

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

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.

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.

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:

$P\left(\stackrel{炉}{X}>240\right)=P\left(\frac{{\displaystyle \frac{x-渭}{蟽}}}{\sqrt{n}}>\frac{{\displaystyle \frac{240-渭}{蟽}}}{\sqrt{n}}\right)$

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

$=P(Z>1.50)\left(Froms\tan dardnormaltable\right)$$=P\left(Z>1.50\right)\left(Froms\tan dardnormaltable\right)$

$=0.0668$

Thus, the required probability is $0.0668$