(18 total points) One of the consequences of the rise of music streaming services is that song lengths have decreased in the last few years. Suppose the length in seconds, X, of a randomly chosen Billboard Hot 100 song can be modeled with a Normal distribution with mean of 210 seconds and standard deviation of 70 seconds. a.

(3 pts) According to the model, the longest 15% of songs are at least how many seconds long?

b. (4 pts) What is the probability that a randomly chosen song from the Hot 100 list lasts longer than 4 minutes?

c. (5 pts) Suppose you randomly choose 4 songs from the Hot 100 list. What is the probability that the average song length is between 2 and 3 minutes?

d. (4 pts) What are the mean and standard deviation of the total length of four randomly chosen Hot 100 songs? Hint: Define a random variable for each of the four songs.

e. (2 pts) In order to calculate the standard deviation in the previous part of the problem, you must make an assumption about the lengths of the four songs. What is that assumption, and what in the description from part c tells you that you can make that assumption?