Question

Below is a cumulative relative frequency graph for the lengths, in minutes, of 200 songs

recorded by the Rolling Stones.

a. According to these data, the mean song length was 4.23 minutes, and the standard
deviation was 1.38 minutes. A music lover who wants to create a mix of songs wants to have 5 seconds of silence between songs, so he needs to add five seconds to the length of each song. He also wants to express the times in seconds, rather than minutes. Find the mean and standard deviation of the transformed data.

b. What are the mean and standard deviation of the z-scores of song lengths? Justify your answer.

Answer 1

Adding 5 seconds would give

Mean= X`= 258.8 seconds

Standard Deviation=S= 82.8 seconds

The standard deviation remains the same.

Mean z= 0

Standard Deviation z= 1

Explanation:

Mean= 4.23 minutes,

Standard Deviation= 1.38 minutes

He wants to add 5 seconds of silence between songs,

So we will convert these minutes into seconds

Mean= 4.23 minutes * 60= 253.8 seconds

Standard Deviation= 1.38 minutes* 60= 82.8 seconds

Adding 5 seconds would give

Mean= X`= 4.23 minutes * 60 +5 = 258.8 seconds

Standard Deviation=S= 1.38 minutes* 60= 82.8 seconds

The standard deviation remains the same.

b. Transforming into z- scores would give

Mean z= x`-u/s= 4.23-4.23/1.38= 0

Standard Deviation z= s/s= 1.38/1.38= 1

This is because changing into z- scores is simply a linear transformation.