Finding the probability and distribution

Question

Suppose that the average number of Facebook friends users have is normally distributed with a mean of 122 and a standard deviation of about 58. Assume forty-four individuals are randomly chosen. Answer the following questions. Round all answers to 4 decimal places where possible.

1. What is the distribution of ?  ~ N(?,?)
2. For the group of 44, find the probability that the average number of friends is less than 132. ?
3. Find the third quartile for the average number of Facebook friends. 
4. For part b), is the assumption that the distribution is normal necessary? 
5. yes or no

Answer 1

distribution of xbar:

xbar ~ N(122,58/sqrt(44)) = N(122,8.7438), where standard error (SE) = 8.7348.

probability average friends < 132:

z = (xbar - mu)/SE

P(xbar < 132) = P(z < (132-122)/8.7438) = (z < 1.14) = 0.8729

3rd quartile = 0.75 probability. z-score for 0.75 = 0.674

0.674 = (xbar - 122)/8.7348

xbar = 127.89

assumption of normality is not necessary because the distribution of sample means is assumed to be normal for large sample sizes (>= 30).