Question

Lack of sleep has been associated with psychological distress in adolescents. In a phone survey from 2013 by the National Sleep Foundation there were about 250 respondents per country aged 25-55 years old. It was found that on average young adults sleep 7.2 hours with standard deviation of 0.8 hour.

1. What is the probability that young adults sleep less than 8.08 hours?

2.

What is the probability that young adults sleep between 6.32 and 8.48 hours?

3. What is the probability that young adults sleep more than 8.48 hours?

4. At least how many hours do young adults sleep who are among the 15 % with the longest hours of sleep?

5. About 11 % of young adults sleep fewer hours than ______

Answer 1

This is a normal distribution question with



1) P(x < 8.08)=?
The z-score at x = 8.08 is,

z = 1.1
This implies that



b) P(6.32 < x < 8.48)=?

This implies that
P(6.32 < x < 8.48) = P(-1.1 < z < 1.6) = P(Z < 1.6) - P(Z < -1.1)
P(6.32 < x < 8.48) = 0.945200708300442 - 0.13566606094638267



3. P(x > 8.48)=?
The z-score at x = 8.48 is,

z = 1.6
This implies that
P(x > 8.48) = P(z > 1.6) = 1 - 0.945200708300442



4) Given in the question
P(X > x) = 0.15
This implies that
P(Z > 1.0364) = 0.85
With the help of formula for z, we can say that

x = 8.0291


5. Given in the question
P(X < x) = 0.11
This implies that
P(Z < -1.2265) = 0.11
With the help of formula for z, we can say that

x = 6.2188
PS: you have to refer z score table to find the final probabilities.
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