The distribution of the number of hours of sleep people get per night is unimodal and symmetric with a mean of 6 hours and a standard deviation of 1.5. Using the Empirical Rule, approximately what percentage of people sleep between 6 and 7.5 hours per night

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Mathematics

Dexter Moreno

Today, 15:09

The distribution of the number of hours of sleep people get per night is unimodal and symmetric with a mean of 6 hours and a standard deviation of 1.5. Using the Empirical Rule, approximately what percentage of people sleep between 6 and 7.5 hours per night

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Ariana Bond · Answer 1

Approximately 34% of people sleep between 6 and 7.5 hours per night

Step-by-step explanation:

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

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

95% of the measures are within 2 standard deviation of the mean.

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

In this problem, we have that:

Mean = 6

Standard deviation = 1.5

The normal distribution is symmetric, which means that 50% of the measures are below the mean and 50% are above.

What percentage of people sleep between 6 and 7.5 hours per night

6 is the mean.

7.5 is 1 one standard deviation above the mean.

By the Empirical Rule, of the 50% of the measures that are above the mean, 68% are within 1 standard deviation of the mean (between 6 and 7.5).

0.5*0.68 = 0.34

Approximately 34% of people sleep between 6 and 7.5 hours per night