Question

The distribution of the number of hours of sleep people get per night is bell shaped with a mean of six hours in standard deviation of 1.5 hours

1. Approximately what percent of people sleep between 6 and 7.5 hours per night?
a. 16%
b. 24%
c. 34%
d. 68%
2. If Aaron had a Z score of 1.2 how many hours did he sleep?
a. 7.5 hours
b. 7.8 hours
c. 8.0 hours
d. 8.2 hours

Answer 1

1. c. 34%

2. b. 7.8 hours

Explanation:

Empirical Rule:

The Empirical Rule states that, for a normally distributed random variable:

Approximately 68% of the measures are within 1 standard deviation of the mean.

Approximately 95% of the measures are within 2 standard deviations of the mean.

Approximately 99.7% of the measures are within 3 standard deviations of the mean.

Z-score:

Problems of normal distributions can be solved using 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:

Mean of 6 hours, standard deviation of 1.5 hours.

1. Approximately what percent of people sleep between 6 and 7.5 hours per night?

6 hours = mean

7.5 hours = 6 + 1.5 = 1 standard deviation above the mean.

Approximately 68% of the measures are within 1 standard deviation of the mean. Since the normal distribution is symmetric, 34% are between one standard deviation below the mean and the mean, and 34% are between the mean(6 hours) and 1 standard deviation above the mean(7.5 hours). So the answer is 34%, given by option c.

2. If Aaron had a Z score of 1.2 how many hours did he sleep?

We have that
`Z = 1.2, \mu = 6, \sigma = 1.5`. We have to find X.

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

`1.2 = (X - 6)/(1.5)`

`X - 6 = 1.2*1.5`

`X = 7.8`

So option b.