Question

The results of a national survey showed that on average, adults sleep 6.9 hours per night. Suppose that the standard deviation is 1.2 hours and that the number of hours of sleep follows a bell-shaped distribution.

a. Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours.

Answer 1

Using Chebyshev's theorem, with a mean of 6.9 hours and a standard deviation of 1.2 hours, at least 75% of adults are expected to sleep between 4.5 and 9.3 hours per night.

Step-by-step explanation:

To calculate the percentage of individuals who sleep between 4.5 and 9.3 hours using Chebyshev's theorem, we can follow these steps:

1. Identify the mean, which is 6.9 hours, and the standard deviation, which is 1.2 hours.
2. Calculate the number of standard deviations that 4.5 and 9.3 hours are away from the mean.
3. 4.5 hours is 2.4 standard deviations away (6.9 - 4.5 = 2.4; 2.4 / 1.2 = 2).
4. 9.3 hours is 2 standard deviations away (9.3 - 6.9 = 2.4; 2.4 / 1.2 = 2).
5. Apply Chebyshev's theorem to find the proportion of the data within these bounds.
6. Since Chebyshev's theorem states that at least (1 - 1/k2) of data from a distribution lies within k standard deviations from the mean, where k is greater than 1.
7. Using k=2, we have at least 1 - 1/22 = 1 - 1/4 = 3/4 or 75% of the data within 2 standard deviations.
8. Therefore, at least 75% of individuals are expected to sleep between 4.5 and 9.3 hours.