The Normal Distribution

Question

z-Scores and Area under the Curve
The National Sleep Foundation recommends that adults between 18−6418-64 years of age sleep between 77 and 99 hours per night. A researcher collected data on the amount of sleep that students in college slept per night. The data were approximately normally distributed with the following mean and standard deviation		mean = 6.86.8 hours	standard deviation = 1.31.3 hours		Use this information and the online normal distribution calculator to answer the questions below. Round your percents to two decimal places.		Note: Use z-scores rounded to two decimal places and the standard normal distribution to avoid roundoff errors.
Normal Calculator
What percent of students slept between 77 hours and 99 hours? %
What percent of students slept less than 7.87.8 hours? %
What percent of students slept more than 8.68.6 hours? %
What percent of students slept less than 3.733.73 hours or more than 10.4310.43 hours ? %

Answer 1

Hi Tra'von,

The percentages you seek are equivalent to probabilities under the normal distribution. Use the classic equation in introductory statistics:

z=(x-mu)/sigma

x=value you are given

mu=mean

sigma=standard deviation

First Question

You are asked about the probability between 7 and 9 hours of sleep, so we need two z-scores. I will designate as z₁ and z₂.

x₁=7

mu=6.868

sigma=1.313

z₁=(7-6.868)/1.313

z₁=0.10

P(Z<0.10)=0.5398

Keep this probability in mind. Let's get z₂.

x₂=9

mu=6.868

sigma=1.313

z=(9-6.868)/1.313

z=1.62

P(Z<1.62)=0.9474

Now, to get the area between the two z-scores, just subtract:

P=0.9474-0.5398

P=0.4076

Second Question

Same equation, only one z this time:

x=7.878

mu=6.868

sigma=1.313

z=(7.878-6.868)/1.313

z=0.77

P(Z<0.77)=0.7794

P=0.7794

Third Question

This is similar to the last one, but we have one additional step at the end. We still compute z.

x=8.686

mu=6.868

sigma=1.313

z=(8.686-6.868)/1.313

z=1.38

P(Z<1.38)=0.9162

Now, the question asked for greater than, so we apply the Complement Rule aka the One Minus Trick:

P(Z>a)=1-P(Z<a)

P(Z>1.38)=1-P(Z<1.38)

P(Z>1.38)=1-0.9162

P=0.0838

Final Question

We need to compute both the probability that a student sleeps less than 3.733 hours and the probability that a student sleeps more than 10.43 hours and add them together. I will again use z₁ and z₂:

x₁=3.733

mu=6.868

sigma=1.313

z₁=(3.733-6.868)/1.313

z₁₌ -2.39

P(Z< -2.39)=0.0084

Keep that probability in mind; let's get z₂:

x₂=10.43

mu=6.868

sigma=1.313

z₂=(10.43-6.868)/1.313

z₂= 2.71

P(Z<2.71)= 0.9966

Remember, though, the question asked for greater than, so we need the 1 minus trick:

P(Z>2.71)=1-P(Z<2.71)

P(Z>2.71)=1-0.9966

P(Z>2.71)=0.0034

Add this to the probability above:

P=0.0034 + 0.0084

P=0.0118

I hope this helps.