a. What is the distribution of ? T - N( b. For the group of 14, find the probability that the average number of friends is more than 94. c. Find the first quartile for the average number of Facebook friends.

Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for you. To get remaining sub-part solved please repost the complete question and mention the sub-parts to be solved.

a.

Central limit theorem:

Based on central limit theorem, if the population is normally distributed then the sampling distribution is also normally distributed regardless of the sample size. If there is no information about the population, for sufficiently large sample size n≥30 sampling distribution of sample mean has the normal distribution.

Let $\overline{x}$be the average number of Facebook friends users which follows normal distribution. The mean is 111 and standard deviation is 59. That is, $\mu =111,\sigma =59$.

The mean of the sampling distribution is,

$\begin{array}{rcl}{\mu }_{\overline{x}}& =& \mu \\ & =& 111\end{array}$

The standard deviation of the sampling distribution is,

$\begin{array}{rcl}{\sigma }_{\overline{x}}& =& \frac{\sigma }{\sqrt{n}}\\ & =& \frac{59}{\sqrt{14}}\\ & =& 15.7684\end{array}$

Thus, the distribution of $\overline{x}$ is $\overline{\mathbf{x}}\mathbf{~}\mathit{N}(111,15.7684)$.