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Suppose my iPod has 3,000 songs.

The histogram below shows a 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.

Calculate the probability that a randomly

selected song lasts more than five minutes.

Here we're looking for the probability

of one randomly selected song lasting more than five minutes.

This is the same thing as saying among all the population of

songs on this iPod, what percentage of them last more than five minutes.

This should be a pretty simple question to calculate.

And lately, what we've been doing was to

calculate Z-score, and use those to find probability.

If that's your instinct here, though, you should not follow it.

Because remember

that we can use Z-scores and the associated normal probabilities

only if the distribution we're working with is nearly normal.

And taking a look at the distribution of songs here, they certainly are not.

The distribution of the lengths of all of

these songs on the iPod is indeed right-skewed.

Does this make sense?

Well, a song can't be less than zero minutes,

so we have a natural boundary at the lower end.

And there's really no

upper end to how long your songs can be.

However, as you can imagine, it's going to be

fewer and fewer songs as the number of minutes increases.

That's what gives us the right skewed distribution here.

So, we've confirmed that the population distribution

makes sense, but we've also said that

the methods that we've learned most recently

for calculating these probabilities don't apply here.

Does this mean we can't answer this question, though?

No.

We can actually use the histogram and the heights of the

bars to estimate what percentage of songs fall between, let's say

four and five minutes, five and six minutes, six and seven,

so on and so forth, and use those to calculate this probability.