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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. a. Use Chebyshev’s theorem to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours. b. Use Chebyshev’s theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 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 4.5 and 9.3 hours per day. How does this result compare to the value that you obtained using Chebyshev’s theorem in part (a)?

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STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884

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STATISTICS F/BUSINESS+ECONOMICS-TE...

13th Edition
Anderson
Publisher: CENGAGE L
ISBN: 9781305881884
Chapter 3.3, Problem 39E
Textbook Problem
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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.

  1. a. Use Chebyshev’s theorem to calculate the percentage of individuals who sleep between 4.5 and 9.3 hours.
  2. b. Use Chebyshev’s theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 hours.
  3. 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 4.5 and 9.3 hours per day. How does this result compare to the value that you obtained using Chebyshev’s theorem in part (a)?

a.

To determine

Find the percentage of individuals who sleep between 4.5 and 9.3 hours by using the Chebyshev’s theorem.

Explanation of Solution

Calculation:

On average, adults sleep 6.9 hours per night and the standard deviation is 1.2 hours.

The formula for z-score is zi=xix¯s.

Where x¯ is the sample mean and s is the sample standard deviation.

The z-score corresponding to 4.5 can be obtained as follows:

Substitute xi=4.5, s=1.2 and x¯=6.9

z=4.56.91.2=2

Thus, the z-score corresponding to 4.5 is –2.

The z-score corresponding to 9.3 can be obtained as follows:

Substitute xi=9.3, s=1.2 and x¯=6.9

z=9

b.

To determine

Find the percentage of individuals who sleep between 3.9 and 9.9 hours, by using the Chebyshev’s theorem.

c.

To determine

Find the percentage of individuals who sleep between 4.5 and 9.3 hours, by using the empirical rule. Compare this result with the result that obtained using Chebyshev’s theorem in part (a)

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