The results of a national survey showed that on average, adults sleep 6.9 hours

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.

Round your answers to the nearest whole number.

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

Use Chebyshev's theorem to calculate the percentage of individuals who sleep between 3.9 and 9.9 hours.
At least  %

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)?
SelectThe empirical rule produces a larger percentage than Chebyshev's theoremChebyshev's theorem produces a larger percentage than the empirical ruleBoth methods produce the same percentage

Explanation / Answer

Part-a

As 4.5 and 9.9 are within 2 standard deviation from mean, so at least 75%of individuals sleep between 3.9 and 9.9 hours

Part-b

As 4.5 and 9.9 are within 2 standard deviation from mean, so at least 95%of individuals sleep between 3.9 and 9.9 hours

Part-c

The empirical rule produces a larger percentage than Chebyshev's theorem

k Min. % within k standard   
deviations of mean Max. % beyond k standard
deviations from mean 1 0% 100% 2 50% 50% 1.5 55.56% 44.44% 2 75% 25% 3 88.8889% 11.1111% 4 93.75% 6.25% 5 96% 4% 6 97.2222% 2.7778% 7 97.9592% 2.0408% 8 98.4375% 1.5625% 9 98.7654% 1.2346% 10 99% 1%

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