Download Section 6

Chapter 7 – The Normal Probability Distribution
This chapter studies continuous random variables. Two such distributions are
covered: Uniform Distribution and Normal Distribution
Read page 372 (green)
Section 7.1 – Properties of the Normal Distribution
Probability Density Function – An equation used to compute probabilities of continuous
random variables. It must satisfy the following 2 properties:
1. Every point on the curve has a vertical height greater than 0.
2. Total area under the curve must equal 1
***The area under the curve = Probability***
Read page 373
Uniform Distribution – A continuous random variable whose values are spread evenly over
a range of possibilities [a, b]. If a histogram for a uniform distribution was created, it would
consist of bars that were all very close in height, indicating that all the outcomes occurred
approximately the same number of times (all possibilities are equally likely).
The model for a uniform distribution is a rectangle. Remember that when we dealt with
discrete probability distributions, we could verify a true distribution by adding up the P(x)
column to make sure that the sum = 1.
With continuous distributions, the probability density graph must be considered to have a
Total Area under the curve to equal 1.
Important note: To find probability we must find an area under the probability density
curve. The area under the graph of a density function over an interval represents the
probability of observing a value of the random variable in the interval.
To find probabilities for a uniform distribution:
a. Draw a rectangle as the model
b. The dimension on the bottom is (b – a)
c. In order to have an area of 1, the height of the rectangle must be 1/(b – a).
d. Shade the portion of the rectangle that represents the probability you want.
e. Find the area of this shaded portion.
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A principal must select a starting time for a fire drill between 8:00am and 12:00pm, and the
selection is made in such a way that all possible times are equally likely. If she randomly
selects a starting time, what is the probability that it is:
a) between 8:00am and 9:00am?
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b) between 10:00am and 10:30am?
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Normal Distribution – A continuous random variable is normally distributed if it is
symmetric and bell-shaped. It is commonly called the “Normal Curve” or “Bell-Shaped
Curve”.
Properties of the Normal Density Curve:
Symmetric about the mean, μ.
Mean = Median = Mode, so there is a single peak and the highest point occurs at x = μ.
Inflection points occur at μ – σ and μ + σ.
The total area under the normal curve is 1.
½ the area is to the right of the mean and ½ the area is to the left of μ .
As x increases and decreases without bound, the graph approaches (but never touches)
the horizontal axis.
7. The shape of the curve is affected by two things: the population mean (μ) and the
population standard deviation (σ).
1.
2.
3.
4.
5.
6.
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8. The Empirical Rule applies:
Approximately 68% of the area under the curve is between μ – σ and μ + σ
Approximately 95% of the area under the curve is between μ – 2σ and μ + 2σ
Approximately 99.7% of the area under the curve is between μ – 3σ and μ + 3σ
(see pg. 376)
#26 page 383 Look at the graph given, what are the values of μ and σ?
The normal curve is a mathematical model (an equation, table, or graph that is used to
describe reality). The normal curve does a good job of describing the distribution of things
like height, IQ scores, birth weights, etc.
The heights of 10-year-old males are normally distributed with a mean of 55.9 inches and a
standard deviation of 5.7 inches.
a) Draw a normal curve with the parameters labeled.
_____________________
b. Shade the region that represents the proportion of 10-year-old males who are less
than 44.5 inches tall.
c. Suppose the area under the normal curve to the left of 44.5 is 0.0496.
Interpret what this means. (see page 378 Example 4 c)
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The standard normal distribution has
μ = 0, σ = 1
If a random variable X is normally distributed with mean μ and standard deviation
x
the random variable Z 
is normally distributed with μ = 0 and σ = 1.

σ, then
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!!
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Section 7.2 – The Standard Normal Distribution
The standard normal distribution is also called a Z distribution, has
μ = 0, σ = 1.
Finding the Area under the Standard Normal Curve:
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 !!
Find the area to the LEFT of a z-score
a.
P (Z < 2.29)
b. P (Z < –1.09)
_____________________
_____________________
Find the area to the RIGHT of a z-score
a. P (Z > 2)
b. P (Z > –1.02)
_____________________
_____________________
Find the area IN BETWEEN two z-scores
a. P (0 < Z < 1.25)
b. P (–0.12 < Z < 1.02)
_____________________
_____________________
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Summary on pg. 389
Calculator:
2nd VARS 2:normalcdf(lowerbound, upperbound, 0,1) ENTER
When there is no lowerbound, enter -1E99. When there is no upperbound, enter 1E99. The
E is scientific notation selected by pressing 2nd then ,.
Finding z-scores when you are given AREA first:
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.
_____________________
b. Find the z-score so that the area to the left of the z-score is 0.9931.
_____________________
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.
_____________________
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b. Find the z-score so that the area to the right of the z-score is 0.9713.
_____________________
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.
_____________________
These problems may say “Find the ________ percentile” or “Find the cutoff value for the
upper _______% of the data” or “Find the value that separates ________” etc.
Find the z-score such that the area under the standard normal curve to its left is 0.25.
This is really asking for what percentile?
Draw a normal curve and shade the given area.
_____________________
Calculator:
To get the z-score when the area to the left of that z-score is known use InvNorm.
2nd VARS 3:InvNorm(area to the left)
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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. Our use of the normal distribution will include other means and standard
deviations. In order to find the area 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
(this converts “x’s” (data points) into “z’s”)
The z-score represents how many standard deviations “x” is from its mean,
x

μ.
Finding the Area under a Normal Curve:
P ( a  X  b)
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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.
#4 page 402 Assume a Normal Distribution with μ= 50 and σ = 7.
Find P(X > 65)
#20 page 402 Wendy’s drive-through times have a mean of 138.5 sec and standard
deviation of 29 sec. Assume normal distribution
a. Find P(drive-through time < 100 sec.)
b. Find P(drive-through time > 160 sec.)
c. What proportion of cars spend between 2 and 3 minutes in the drive-through lane?
d. Would it be unusual for a car to spend more than 3 minutes in Wendy’s drive-through?
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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?
Finding the Value of a Normal Random Variable that Corresponds to a Given
Proportion, Probability, or Percentile:
1. Draw a standard normal curve and shade the area corresponding to the given
probability or percentile.
2. Find the Z-score that corresponds to the shaded area.
Use invNorm(area to the left) to find the appropriate z-score.
3. Convert the Z-score back to a data point using the formula x    z .
#16 page 402
#26 page 403
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)?
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.
Calculator:
2nd VARS 2:normalcdf(lowerbound, upperbound, μ, σ) ENTER
When there is no lowerbound, enter -1E99. When there is no upperbound, enter 1E99. The
E is scientific notation selected by pressing 2nd then ,.
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