Question

The results of a national survey showed that on average, adults sleep 6.7 hours per night. Suppose that the standard deviation is 1.6 hours. (a) Use Chebyshev's theorem to calculate the minimum percentage of individuals who sleep between 3.5 and 9.9 hours. % (b) Use Chebyshev's theorem to calculate the minimum percentage of individuals who sleep between 2.7 and 10.7 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 3.5 and 9.9 hours per day. % How does this result compare to the value that you obtained using Chebyshev's theorem in part (a)? The percentage obtained using the empirical rule is the percentage obtained using Chebyshev's theorem.

Answer 1

Answer:

a) Atleast 75%

b) Atleast 88.9%

c) 95%

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 6.7

Standard Deviation, σ = 1.6

Chebyshev's Theorem:

• Atleast $1 - \dfrac{1}{k^2}$ percent of data lies within k standard deviation of mean.

Empirical Rule:

• Almost all the data lies within 3 standard deviation for a normal data.
• Around 68% of data lies within 1 standard deviation of mean.
• Around 95% of data lies within 2 standard deviation of mean.
• Around 99.7% of data lies within 3 standard deviations o mean.

a) minimum percentage of individuals who sleep between 3.5 and 9.9 hours

$3.5 = \mu - 2\sigma = 6.7 - 2(1.6)\\9.9 = \mu + 2\sigma = 6.7 + 2(1.6)$

We have to find the percent of data lying within 2 standard deviation of mean.

Putting k =  2

$1 - \dfrac{1}{(2)^2} = 0.75 = 75\%$

Atleast 75% of of individuals who sleep between 3.5 and 9.9 hours.

b) minimum percentage of individuals who sleep between 2.7 and 10.7 hours

$2.7 = \mu - 3\sigma = 6.7 - 3(1.6)\\10.7 = \mu + 3\sigma = 6.7 + 3(1.6)$

We have to find the percent of data lying within 3 standard deviation of mean.

Putting k =  3

$1 - \dfrac{1}{(3)^2} = 0.889 = 88.9\%$

Atleast 88.9% of of individuals who sleep between 2.7 and 10.7 hours.

c)  percentage of individuals who sleep between 3.5 and 9.9 hours per day.

Thus, we have to find the percentage of data lying within 2 standard deviation.

By empirical rule, around 95% of data lies within 2 standard deviation.

Thus, 95% of of individuals who sleep between 3.5 and 9.9 hours.

This is higher than the percentage obtained from Chebyshev's rule.