Question

The length of songs in a collector’s iTunes album collection is uniformly distributed from two to 3.5 minutes. Suppose we randomly pick five albums from the collection. There are a total of 43 songs on the five albums a) Find the probability that 43 songs will last on average more than 2.78 minutes b) Find the probability that on average 43 songs will last less than 2.8 mnutes

Answer 1

Uniform distribution from a = 2 minutes to b = 3.5 minutes

Mean, = (a + b)/2

= (2 + 3.5)/2

= 2.75 minutes

Standard deviation, =

=

= 0.433 minutes

For sampling distribution of mean with sample size n,

P( < A) = P(Z < )/), where is the sample mean and is the sample standard deviation (standard error)

n = 43

= = 2.75 minutes

=

=

= 0.066

a) P(43 songs will last on average more than 2.78 minutes) = P( > 2.78)

= 1 - P( < 2.78)

= 1 - P(Z < (2.78 - 2.75)/0.066)

= 1 - P(Z < 0.45)

= 1 - 0.6736

= 0.3264

b) P(on average 43 songs will last less than 2.8 minutes) = P( < 2.8)

= P(Z < (2.8 - 2.75)/0.06)

= P(Z < 0.76)

= 0.7764