I need help with writing distribution of x̅, finding average of 1st quartile, and finding IQR

Question

1. The length of songs in a collector's iTunes album collection is uniformly distributed from two to 3.7 minutes. Suppose we randomly pick five albums from the collection. There are a total of 42 songs on the five albums.

I need help writing the distribution of x̅, finding the first quartile for the average song length, and finding the IQR for the average song length.

Please explain, don't just give me the answer, I want to be able to understand the steps you did to solve it

Answer 1

We use the Central Limit Theorem. Since the Uniform distribution has finite variance, and we have independent samples, it meets the CLT conditions.

For X ~ U[2, 3.7], µ = (2 + 3.7)/2 = 2.85 or 2 17/20, var = (3.7 - 2)²/12 = 289/1200

Then, for the sample with n=12, µ = 2.85 and var = 289/1200/42 = 289/50400, and σ = √var = 17√14/840 =

0.0757240185418536

The Central Limit Theorem tells us that the distribution approaches the normal distribution. That the original distribution is the uniform and n is relatively large assures us that it will be "close", and we are looking at quartiles, so all should be well.

Then, the distribution of the sample is approximately N(2 17/20, 289/50400) or

N(2.85, 0.00573412698412698)

Then, for the first quartile, x_.25 = µ + z_.25 σ = 2.85 + normsinv(.25) * 0.0757240185418536 =

2.79892492564986

From symmetry of the normal, The IQR (interquartile range) equals 2 * z_.75 σ = 2 * normsinv(.75) * 0.0757240185418536 = 0.102150148700277