Download Section 7.5 – The Normal Distribution Section 7.6 – Applications of

Section 7.5 – The Normal Distribution
Section 7.6 – Application of the Normal Distribution
A random variable that may take on infinitely many values is called a continuous
random variable.
A continuous probability distribution is defined by a function f called the probability
density function.
The probability that the random variable X associated with a given probability density
function assumes a value in an interval a < x < b is given by the area of the region
between the graph of f and the x-axis from x = a to x = b.
The following graph is a picture of a normal curve and the shaded region is
P(a < X < b).
Note: P(a < X < b) = P(a < X < b) = P(a < X < b) = P(a < X < b), since the area under
one point is 0. The area of the region under the standard normal curve to the left of
some value z, i.e. P(Z < z) or P(Z  z), is calculated for us in the Standard Normal
Cumulative Probability Table found in Chapter 7 of the online book.
Section 7.5 – The Normal Distribution
Section 7.6 – Applications of the Normal Distribution
1
Normal distributions have the following characteristics:
1. The graph is a bell-shaped curve.
The curve always lies above the x-axis but approaches the x-axis as x extends indefinitely
in either direction.
2. The curve has peak at x =  . The mean,  , determines where the center of the curve
is located.
3. The curve is symmetric with respect to the vertical line x =  .
4. The area under the curve is 1.
5.  determines the sharpness or the flatness of the curve.
6. For any normal curve, 68.27% of the area under the curve lies within 1 standard
deviation of the mean (i.e. between    and    ), 95.45% of the area lies within 2
standard deviations of the mean, and 99.73% of the area lies within 3 standard deviations
of the mean.
The Standard Normal Variable will commonly be denoted Z. The Standard Normal
Curve has  =0 and  =1.
Example 1: Let Z be the standard normal variable. Find the values of:
a. P(Z < -1.91)
Section 7.5 – The Normal Distribution
Section 7.6 – Applications of the Normal Distribution
2
b. P(Z > 0.5)
c. P(-1.65 < Z < 2.02)
Example 2: Let Z be the standard normal variable. Find the value of z if z satisfies:
a. P(Z < -z) = 0.9495
b. P(Z > z) = 0.9115
c. P(-z < Z < z) = 0.8444
1
Formula: P ( Z  z )  1  P( z  Z  z )
2
Section 7.5 – The Normal Distribution
Section 7.6 – Applications of the Normal Distribution
3
When given a normal distribution in which   0 and   1 , we can transform the
normal curve to the standard normal curve by doing whichever of the following applies.
b

P(X < b) = P Z 

 

a

P(X > a) = P Z 

 

b
a
Z
P(a < X < b) = P

 
 
Example 3: Suppose X is a normal variable with  = 7 and   4 . Find P(X > -1.35).
Applications of the Normal Distribution
Example 4: The heights of a certain species of plant are normally distributed with a mean
of 20 cm and standard deviation of 4 cm. What is the probability that a plant chosen at
random will be between 10 and 33 cm tall?
Section 7.5 – The Normal Distribution
Section 7.6 – Applications of the Normal Distribution
4
Example 5: Reaction time is normally distributed with a mean of 0.7 second and a
standard deviation of 0.1 second. Find the probability that an individual selected at
random has a reaction time of less than 0.6 second.
Approximating the Binomial Distribution Using the Normal Distribution
Theorem
Suppose we are given a binomial distribution associated with a binomial experiment
involving n trials, each with probability of success p and probability of failure q. Then if
n is large and p is not close to 0 or 1, the binomial distribution may be approximated by a
normal distribution with   np and   npq .
Example 6: Consider the following binomial experiment. Use the normal distribution to
approximate the binomial distribution. A company claims that 42% of the households in
a certain community use their Squeaky Clean All Purpose cleaner. What is the
probability that between 15 and 28, inclusive, households out of 50 households use the
cleaner?
Section 7.5 – The Normal Distribution
Section 7.6 – Applications of the Normal Distribution
5
Example 7: Use the normal distribution to approximate the binomial distribution. A
basketball player has a 75% chance of making a free throw. She will make 120 attempts.
What is the probability of her making:
a. 100 or more free throws?
b. fewer than 75 free throws?
Example 8: Use the normal distribution to approximate the binomial distribution. A fair
coin is tossed 20 times. What is the probability of obtaining the following
number of heads:
a. Fewer than 8 heads?
b. More than 6 heads?
Section 7.5 – The Normal Distribution
Section 7.6 – Applications of the Normal Distribution
6
M1313 popper number 21
Use the following information for questions 1, 2 and 3
x
P(X= x)
1
2
4
.15
.45
.40
1. Find the expected value.
A. 2.39
B. 3
The expected value is 2.65
C. 2.5
D. 2.65
2. Find the variance.
A. 1.33
3.
B. 1.08
D. 1.03
Find the standard deviation.
A. 1
4.
C. 1.42
B. 1.15
C. 1.19
D. 1.01
Given
−3
x
P(X =x)
0
.5
3
.2
.3
Find the variance given that the expected value is − .6
A. 6.84
5.
B. 2.85
C. 8.142
D. 2.02
A probability distribution has a mean of 10 and standard deviation
of 1.5. Use Chevbychev’s Inequality to estimate the probability that
an outcome will be between 7 and 13.
A.
1
4
B.
1
2
C.
3
5
D.
3
4
M 1313 Popper number 19
Use for questions 1, 2 and 3.
An automobile manufacturer obtains the microprocessors used to regulate
fuel consumption in its automobiles from three microelectronic firms: A, B,
and C. The quality-control department of the company has determined
that 1% of the microprocessors produced by firm A are defective, 2% of
those produced by firm B are defective, and 1.5% of those produced by
firm C are defective. Firms A, B, and C supply 45%, 25%, and 30%,
respectively, of those microprocessors used by the company. An
automobile is selected at random. Draw a tree diagram.
1. What is the probability it was manufactured at firm B and it was found
to be defective?
A. .0050
2.
C. .0140
D. .0200
What is the probability it is defective?
A. .0038
3.
B. .0150
B. .0150
C. .0140
D. .0200
What is the probability it was defective given it was manufactured at
firm C?
A. .0038
B. .0150
C. .0140
D. .0200
Section 20042
Assignment 015
Grade ID
Form A
Use for questions 4 and 5.
An urn contains 10 red and 13 blue marbles. Two marbles are chosen at
random, in succession and without replacement. Draw tree diagram.
4. What is the probability a marble is blue?
A. .4348
B. .5652
C. .5504
D. .5455
5. What is the probability that the first marble was red, given that the
second one was blue?
A. .4546
B. .5652
C. .5504
D. .5455