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COLLIN COLLEGE
Chapter 7
THE NORMAL PROBABILITY DISTRIBUTION
Math 1342.S06
Prof. Ntchobo
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Section 7.1 – Properties of the Normal Distribution
Probability Density Function
The area under the graph of the density function over an interval represents the probability of
observing a value of the random variable in that interval.
Normal Distribution – A continuous random variable is normally distributed if it is symmetric and
bell-shaped.
Properties of the Normal Density Curve:
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Example: The weights of giraffes are approximately normally distributed with mean μ = 2200 pounds
and standard deviation σ = 200 pounds.
a) Draw a normal curve with the parameters labeled.
b) Shade the area under the normal curve to the left of x = 2100 pounds.
c) Suppose that the area under the normal curve to the left of x = 2100 pounds is 0.3085.
Provide two interpretations of this result.
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If a normal random variable has a mean different from 0 or a standard deviation different from 1, we
can convert the normal random variable into a standard normal random variable, Z, and then use ONE
table to calculate the area (probability).
This is extremely important!!
Example: The weights of giraffes are approximately normally distributed with mean μ = 2200 pounds
and standard deviation σ = 200 pounds. Draw a graph that demonstrates the area under the normal
curve between 2000 and 2300 pounds is equal to the area under the standard normal curve between the
Z-scores of 2000 and 2300 pounds.
Section 7.2 – The Standard Normal Distribution
Finding the Area under the Standard Normal Curve:
The table gives the area under the standard normal curve for values to the left of a specified Z-score, z ,
o
as shown in the figure.
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Find the area under the standard normal curve to the left of z = -0.38.
z-score: The distance along the horizontal scale of the normal distribution. It counts how many
standard deviations above or below the mean a particular data point is. ALWAYS 2 DECIMALS !!
Area: The region underneath the curve. It is probability!! ALWAYS 4 DECIMALS !!
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Example: Finding area to the LEFT of a z-score
a)
P (Z < 2.29)
b) P (Z < –1.09)
Example: Finding area to the RIGHT of a z-score
a) P (Z > 2)
b) P (Z > –1.02)
Example: Finding area IN BETWEEN two z-scores
a) P (0 < Z < 1.25)
b) P (–0.12 < Z < 1.02)
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Finding z-scores when you are given AREA first:
Find the z-score such that the area to the left of the z-score is 0.7157.
Find the z-scores that separate the middle 80% of the area under the normal curve from the 20% in the
tails.
Example: Finding a z-score from an area to the LEFT of the z-score:
a) Find the z-score so that the area to the left of the z-score is 0.0262.
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b) Find the z-score so that the area to the left of the z-score is 0.9931.
Example: Finding a z-score from an area to the RIGHT of the z-score:
a) Find the z-score so that the area to the right of the z-score is 0.4325.
b) Find the z-score so that the area to the right of the z-score is 0.9713.
Example: Finding a z-score from an area IN BETWEEN two z-scores:
a) Find the z-score that separates the middle 70% of the area under the standard normal curve
from the area in the tails.
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b) Find the z-score that separates the middle 20% of the area under the standard normal curve
from the area in the tails.
Example: Find the value of b. Assume that the z-scores are normally distributed with a mean of 0 and
a standard deviation of 1.
a) P(–b < Z < b) = 0.9742
b) P(Z > b) = 0.2236
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Section 7.3 – Applications of the Normal Distribution
In this section, we will be working with problems that come from non-standard normal distributions.
In order to find the are we will need to convert the normal random variables values into standardized
normal random variables.
To convert a non-standard value to a standardized z-score, use:
z
x

(this converts “x’s” (data points) into “z’s”)
Finding the Area under a Normal Curve:
P ( a  X  b)
1.
2.
3.
4.

b
a
P
Z
 
 
Write the inequality statement for X.
Convert the values of X into Z-scores.
Draw a standard normal curve and shade the desired area.
Find the area under the curve.
Example: Women’s heights are normally distributed with μ = 63.6 inches and σ = 2.5 inches.
Find the probability that a randomly selected woman will be:
a) taller than 62 inches
b) between 63.6 and 71 inches tall
Example: The scores for males on the math portion of the SAT test are normally distributed with a
mean of 531 and a standard deviation of 114. If a particular college requires a minimum
score of 600, what percentage of males do not satisfy the requirement?
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Finding the Value of a Normal Random Variable that Corresponds to a Given Proportion,
Probability, or Percentile:
Example: Women’s heights are normally distributed with μ = 63.6 inches and σ = 2.5 inches.
Suppose that a modeling agency will only accept the tallest 10% of women. What is the cutoff
height that the agency uses (ie, how tall would a woman have to be to be hired by the agency)?
Example: I.Q. scores are normally distributed with a mean of 100 and a standard deviation of 15.
a) People are considered “intellectually very superior” if their score is above 130.
What percentage of people fall into that category?
b) If we redefine the category of “intellectually very superior” to be scores in the top 1%,
what does the minimum score become?
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Example: The lengths of human pregnancies are approximately normally distributed with a mean of
266 days and a standard deviation of 16 days.
a) In a letter to “Dear Abby”, 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 the result suggest?
b) If we stipulate that a baby is premature if the length of the pregnancy is in the lowest 4%,
find the length (days) that separates premature babies from those who are not premature.
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