Question

The average adult gets 7.45 hours of sleep each night, with a standard deviation of 0.65 hours. A pharmaceutical company developing a sleep aid is researching how much sleep the top 1% of adults get each night, on average. Use a calculator to find how many hours of sleep must an adult get each night to be in the top 1% if the company is only basing their initial research on the sleep habits of 30 adults.

Answer 1

2.31

.12

7.73

Explanation:

σx¯=0.653–√0=0.12

By plugging all the numbers into the formula z=x¯−μσx¯ we find that

2.31=x¯−7.450.12

0.28=x¯−7.45

7.73=x¯

Answer 2

At least 8.96 hours of sleep to be in the top 1%.

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 7.45, \sigma = 0.65`

How many hours of sleep to be on the top 1%?

The top 1% is the 100 - 1 = 99th percentile, which is X when Z has a pvalue of 0.99. So X when Z = 2.327. Then

`Z = (X - \mu)/(\sigma)`

`2.327 = (X - 7.45)/(0.65)`

`X - 7.45 = 0.65*2.327`

`X = 8.96`

At least 8.96 hours of sleep to be in the top 1%.