Question

The length of songs in a collector's iTunes album collection is uniformly distributed from two to 3.1 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums.

1. Give the distribution of X. (Round your answers to four decimal places.)

2. Find the first quartile for the average song length. (Round your answer to two decimal places.)

3. Find the IQR (interquartile range) for the average song length. (Round your answer to two decimal places.)

Answer 1

X be the random variable of the length of songs in a collection in minutes.

1. Distribution of X

X has uniform distribution with the smallest parameter a = 2 minutes and the largest parameter b = 3.1 minutes.

2. First quartile

n = 43 songs.

The mean and standard deviation of the uniform distribution is,

Here sample size is 43 that is more than 30, according to the Central limit theorem the distribution of average length of the song is approximately normal with a mean and standard deviation

The mean and standard deviation of xbar are,

The first quartile means Q!, for that, there is 25% of data fall below it means P(Xbar < Q1) = 0.25

First, find the z score for the below area 0.25

Using the standard normal table the z score for the area 0.25 is -0.67

That is z = -0.67

The formula of z score is,

This formula can be rearranged in terms of Q1 as,

First quartile for the average length of song = 2.52 minutes

3. Interquartile range (IQR)

The formula of the interquartile range is,

IQR = Q3 - Q1

Where Q1 is the first quartile and Q3 is the third quartile.

Q1 is 2.52.

For Q3 there is 75% data below it that is P(Xbar < Q3) = 0.75

First, find the z score for the below area 0.75

Using the standard normal table the z score for the area 0.75 is 0.67

That is z = 0.67

The formula of z score is,

This formula can be rearranged in terms of Q3 as,

Third quartile for the average length of song = 2.58 minutes

IQR = Q3 - Q1 = 2.58 - 2.52 = 0.06

The IQR (interquartile range) for the average song length is 0.06 minutes.