Question

4.41 Songs on an iPod: Suppose an iPod has 3,000 songs. The histogram below shows the
distribution of the lengths of these songs. We also know that, for this iPod, the mean length is 3.45
minutes and the standard deviation is 1.63 minutes.

(a) Calculate the probability that a randomly selected song lasts more than 5 minutes (notice that
the distribution is skewed).

Answer 1

The probability that the song lasts more than 5 minutes is 17.1%

Z score is used to determine by how many standard deviations the raw score is above or below the mean. It is given by:

$z=\frac{x-\mu}{\sigma} \\\\where\ x=raw\ score, \mu=mean,\sigma=standard\ deviation$

Given that Mean = 3.45 minutes, standard deviation = 1.63 minutes

For x > 5 minutes:

$z=\frac{x-\mu}{\sigma} \\\\z=\frac{5-3.45}{1.63} \\\\z=0.95$

From the normal distribution table, P(x > 5) = P(z > 0.95) = 1 - P(z < 0.95) = 1 - 0.8289 = 17.1%

The probability that the song lasts more than 5 minutes is 17.1%

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