Transcript Applications of the Normal Distribution

Finding Probability Using the Normal
Curve
Section 6.3
Objectives
 Calculate probability using normal distribution
Key Concept
 This section presents methods for working with normal
distributions that are not standard (NON-STANDARD).
That is the mean, m, is not 0 or the standard deviation, s is
not 1 or both.
 The key concept is that we transform the original variable, x,
to a standard normal distribution by using the following
formula:
Conversion Formula
original value  mean x  m
z

s tan dard deviation
s
Round z  scores to 2 decimal places
Converting to Standard Normal
Distribution
z=
x-m
s
P
P
(a)
m
x
(b)
0
z
Cautions!!!!
 Choose the correct (left/right) of the graph
 Negative z-score implies it is located to the left of the mean
 Positive z-score implies it is located to the right of the mean
 Area less than 50% is to the left, while area more than 50% is to the
right
 Areas (or probabilities) are positive or zero values, but they are
never negative
Example
 According to the American College Test (ACT), results from
the 2004 ACT testing found that students had a mean reading
score of 21.3 with a standard deviation of 6.0. Assuming that
the scores are normally distributed:
 Find the probability that a randomly selected student has a
reading ACT score less than 20
 Find the probability that a randomly selected student has a
reading ACT score between 18 and 24
 Find the probability that a randomly selected student has a
reading ACT score greater than 30
Example
 Women’s heights are normally distributed with a mean 63.6
inches and standard deviation 2.5 inches. The US Army
requires women’s heights to be between 58 inches and 80
inches. Find the percentage of women meeting that height
requirement. Are many women being denied the
opportunity to join the Army because they are too short or
too tall?
Find z-Values Using the Normal Curve
Section 6.4
Example
 According to the American College Test (ACT), results from
the 2004 ACT testing found that students had a mean reading
score of 21.3 with a standard deviation of 6.0. Assuming that
the scores are normally distributed:
 Find the 75th percentile for the ACT reading scores
Example
 The lengths of pregnancies are normally distributed with a mean
of 268 days and a standard deviation of 15 days.
 One classical use of the normal distribution is inspired by a letter to
“Dear Abby” in which a wife claimed to have given birth 308 days
after a brief visit from her husband, who was serving in the Navy.
Given this information, find the probability of a pregnancy lasting 308
days or longer. What does this result suggest?
 If we stipulate that a baby is premature if the length of the pregnancy is
in the lowest 4%, find the length that separates premature babies
from those who are not premature. Premature babies often require
special care, and this result could be helpful to hospital administrators
in planning for that care
Example
 Men’s heights are normally distributed with a mean of 69.0
inches and standard deviation of 2.8 inches.
 The standard casket has an inside length of 78 inches
 What percentage of men are too tall to fit in a standard casket?
 A manufacturer of caskets wants to reduce production costs by
making smaller caskets. What inside length would fit all men
except the tallest 1%?
Assignment
 Page 270 #1-7 odd
 Page 279 #19-25 odd