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MATH 221
Integrated Learning System
Week 3, Lecture 2
Normal Distribution: Finding
Probabilities and Values
Comparing Normal
Distributions
The Scholastic Aptitude Test (SAT) and the ACT are
exams used by colleges to evaluate undergraduate
applicants. SAT scores are normally distributed with a
mean of 1000 and a standard deviation of 200. ACT
scores are normally distributed with a mean of 20 and a
standard deviation of 5. Assume a student took both
tests and scored 1225 on the SAT and 26 on the ACT.
Which is better? How do you know?
Probability and the Normal
Distribution
The lengths of Atlantic croaker fish are normally
distributed with a mean of 10 inches and a standard
deviation of 2 inches. You are fishing on a pier on the
east coast of the United States and catch a croaker.
What is the probability that the croaker is exactly 10
inches long?
(Adapted from problem 8, page 222 in Larson and
Farber)
Probability and the Normal
Distribution
Distribution of Atlantic Croaker Fish
5
6
7
8
9
10
11
Length (inches)
12
13
14
15
Probability and the Normal
Distribution
Distribution of Atlantic Croaker Fish
0.5
0.30854
5
6
7
8
9
10
11
Length (inches)
12
13
14
15
Standard Normal Probability
Table
Using the Table
Using Technology: TI-83
normalpdf(x,,) – Probability density function
normalcdf(L,U,,) – Cumulative probability density
function
Using Technology: EXCEL
Using Technology: EXCEL
Exercise 1
In a recent year the ACT scores for high school students
with a 3.50 to 4.00 grade point average were normally
distributed with a mean of 24.3 and a standard
deviation of 4.2. A student who took the ACT is
randomly selected.
1. Find the probability that the student’s ACT score is
less than 20.
2. Find the probability that the student’s ACT score is
between 20 and 29.
3. Find the probability that the student’s ACT score is
greater than 29.
Using Technology: TI-83
norminv(p) – inverse standard normal function
Finding Values
x x
z 

z  x  x
x  x  z
Exercise 2
In a survey of men in the U.S. (ages 20 – 29), the mean
height was 69.2 inches with a standard deviation of
2.9 inches.
1. What height represents the 95th percentile?
2. What height represents the first quartile?