Adapted from the forthcoming CyberStats introductory statistics course.

Page 1 - Solving Normal Curve Problems

On this page, we'll demonstrate how to solve normal curve problems.

As an example, we'll use the normal curve model for college students' hours of sleep on a week night that we described in 'The Basics' section.  In this model, the mean, m, is 7 hours and the standard deviation, s, is 1.7 hours.

Two Useful Pointers

For any normal curve problem

• It helps to draw a picture of the problem.
• The problem can be converted to a question about standardized scores.

How to Find the Proportion That's Less Than a Value

To find the proportion of a population with a value less than a specified value -

1. Calculate a Z score for the specified value.
2. Use the calculator to determine Prob<Z.

Interactivity

About what proportion of students sleep less than 5 hours on a week night?   The solution - Calculate the standardized score for 5 hours of sleep. The formula is . For 5 hours, Z = ( 5 - 7 ) / 1.7 = -2 / 1.7 = -1.18 For Z = - 1.18, use the calculator to determine Prob<Z. Put -1.18 under Z Score, click Z Score Prob<Z The answer is 0.1190 (about 12% ). About 12% of college students sleep less than 5 hours on a week night.

How to Find the Proportion that's Greater than a Value

To find the proportion of a population with a value greater than a specified value:

1. Calculate a Z score for the specified value.
2. Use the calculator to determine Prob<Z.
3. Calculate 1 - Prob<Z

Interactivity

About what proportion of students sleep more than 10 hours on a week night? The solution - Calculate the Z score for 10 hours. Z = ( 10 - 7 ) / 1.7 = 3 / 1.7 = 1.76 For Z =1.76, click Calc P. Z Score Prob<Z Prob<1.76 is 0.9608 ( about 96% ) This is the proportion sleeping less than 10 hours.  The answer = 1 - 0.9608 = about 0.04 (about 4%). About 4% of college students sleep more than 10 hours on a week night.

How to Find the Proportion in an Interval

To find the proportion of a population that falls in a specified interval -

1. Determine the percentile rank of each value.
2. Calculate the difference between the two percentile ranks.

Example

On a week night, what percentage of students sleep between 5 and 10 hours?   The solution - We learned information about 5 and 10 hours of sleep in the previous two solutions. About 12% sleep less than 5 hours. About 96% sleep less than 10 hours   The answer is 96% - 12% = 84%, the difference between the two percentile ranks. About 84% of college students sleep between 5 and 10 hours on a week night.

How to Find the Value That Has a Given Percentile Rank

To find the value corresponding to a specified percentile rank -

1. Determine the Z score that has the given percentile rank.
2. Figure out what value has that Z score.

Interactivity

What is the 98th percentile of the distribution of hours of sleep? The solution - 1.  Find the Z score that has Prob<Z = 0.98. Use the calculator, put 0.98 under Prob<Z, click Calc Z. Z Score Prob<Z The value under Z Score is 2.05. The 98th percentile is 2.05 standard deviations above the mean.  2. Determine the sleep value that has a Z score of 2.05.   2.05 standard deviations is ( 2.05 )( 1.7 )    = 3.5 hours. 3.5 hours above the mean of 7 hours    = 10.5 hours. The 98th percentile of the sleep distribution is 10.5 hours.

A formula to calculate the value that has a known Z score is

For Z = 2.05 , x = (2.05) (1.7) + 7 = 3.5 + 7 = 10.5 hours.

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