Transcript Applications of the Normal Distribution

Applications of the Normal
Distribution
Section 6.4
Objectives
 Find the probabilities for a normally distributed variable by
transforming it into a standard normal variable
 Find specific data values given the percentages, using the
standard 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!!!!
 Don’t confuse z-scores and
areas
 Z-scores are distances on the
horizontal scale. Table E lists
z-scores in the left column
and across the top row
 Areas (probabilities or
percentages) are regions
UNDER the normal curve.
Table E lists areas in the body
of the table
 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
Finding Areas given a specified
variable, x
 Draw a normal distribution curve labeling the mean and x
 Shade the area desired
 Convert x to standard normal distribution using z-score
formula
 Use procedures on page 287-288
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?
Finding variable x given a specific
probability
 Draw a normal distribution labeling the given probability
(percentage) under the curve
 Using Table E, find the closest area (probability) to the given
and then identify the corresponding z-score (WATCH
SIGN!!!!)
 Substitute z, mean, and standard deviation in z-score formula
and solve for x.
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 312 #1-27 odd