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6.4 Applications of the Normal Distribution
Objective: Find probabilities for a normally distributed variable by transforming
it into a standard normal variable.
Requirement to use the Standard Normal Distribution Curve
 The variable must be normally or approximately normally distributed
In Chapter 6, assume the distributions meet this requirement.
Using the Standard Normal Distribution Curve to Solve Problems
Method
1. Transform the normal distribution into a STANDARD Normal Distribution using the
formula:
This process transforms the variable into z values.
2. Use the z value and area under the curve to solve the problem
Solve the following using the Standard Normal Distribution process
Example 1 If students’ scores for a test have a mean of 100 and a standard deviation of
15, find the percentage of scores that will fall below 112.
Example 2 Each month, an American household generates an average of 28 pounds of
newspaper for garbage recycling. Assume the standard deviation is two pounds. If a
household is selected at random, find the probability of its generating between 27 and 31
pounds per month.
Example 3 Using the scenario described in #1 above, find the probability of a
household generating more than 30.2 pounds per month.
Example 4 The American Automobile Association reports that the average time it takes to
respond to an emergency call is 25 minutes. Assume the variable is approximately normally
distributed and the standard deviation is 4.5 minutes. If 80 calls are randomly selected,
approximately how many will be responded to in less than 15 minutes?
1
NOTE: When doing percentage problems, change the percentage to a decimal before
multiplying and round the answer to the nearest whole number. Most items like this can’t
be divided into parts.
6.4 Application of the Normal Distribution (Day 2)
Finding Data Values Given Specific Probabilities
Example 5 In order to qualify for a police academy, candidates must score in the top 10% on
a general abilities test. The test has a mean of 200 and a standard deviation of 20.
Find the lowest possible score to qualify. Assume test scores are normally distributed.
You can also use an alternative formula when looking for the value of the variable X
X=z∙σ+μ
Example 5 For a medical study, a researcher wishes to select people in the middle 60% of the
population based on blood pressure. If the mean systolic blood pressure is 120 and the
standard deviation is 8, find the upper and lower readings that would qualify people to
participate in the study.
Determining Normality
Method 1 Draw a histogram and determine if it looks normal
Method 2 Use Pearson’s Index of Skewness, PI
If -1 < PI < 1, the distribution is approximately normal
PI 
3( X  median)
s
s – standard deviation
Method 3 Check for outliers. If a value is more than 1.5(Q3 – Q1) below Q1 or
1.5(Q3– Q1) above Q3, then the distribution is skewed.
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Example 6 A survey of 18 high-technology firms showed that the number of days’
inventory they had on hand. Determine if the data are approximately normally
distributed.
5
81
29
88
34
91
44
97
45
98
63
113
68
118
74
151
74
158
Example 7 The data shown consist of the number of games played each year in the career
of Baseball Hall of Famer Bill Mazeroski. Determine if the data are approximately
normally distributed.
81
159
163
148
142
143
152
34
67
135
162
112
151
130
70
152
162
3