The results of a national survey showed that on average, adults sleep 6.7 hours per night. Suppose that the standard deviation is 1.8 hours. (a) Use Chebyshev's theorem to calculate the minimum percentage of individuals who sleep between 3.1 and 10.3 hours. % (b) Use Chebyshev's theorem to calculate the minimum percentage of individuals who sleep between 2.2 and 11.2 hours. % (c) Assume that the number of hours of sleep follows a bell-shaped distribution. Use the empirical rule to calculate the percentage of individuals who sleep between 3.1 and 10.3 hours per day. % How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)

According to this rule almost all the data lies within three standard deviation of the mean for a normally distributed data.
68% of the data lies within one standard deviation of the mean.
About 95% of the data lies within two standard deviation from the mean.
99.7% of the data lies within three standard deviation of the mean.

a) minimum percentage of individuals who sleep between 3.1 and 10.3 hours

[tex]10.3 = 6.7 + 2(3.1) = \mu + 2(\sigma)\\3.1 = 6.7 - 2(3.1) = \mu - 2(\sigma)[/tex]

Minimum percentage:

[tex]1-\dfrac{1}{4} = 75%[/tex]

Thus, minimum 75% of individuals who sleep between 3.1 and 10.3 hours.

b) minimum percentage of individuals who sleep between 2.2 and 11.2 hours.

[tex]11.2 = 6.7 + 2.5(3.1) = \mu + 2.5(\sigma)\\2.2 = 6.7 - 2.5(3.1) = \mu - 2.5(\sigma)[/tex]

Minimum percentage:

[tex]1-\dfrac{1}{(2.5)^2} = 84\%[/tex]

Thus, minimum 84% of individuals who sleep between 2.2 and 11.2 hours.

c) percentage of individuals who sleep between 3.1 and 10.3 hours per day

According to Empirical rule about 95% of the data lies within 2 standard deviations from the mean.

Thus, 95% of individuals who sleep between 3.1 and 10.3 hours.

d) Comparison with Chebyshev's theorem

This is greater than the results obtained from the Chebyshev's theorem in part (a)