Download Chapter 7.2 Applications of the Normal Distribution

Chapter 7.1 Uniform and Normal Distribution
Objective A : Uniform Distribution
A1. Introduction
Recall: Discrete random variable probability distribution
Special case: Binomial distribution
Finding the probability of obtaining a success in n independent trials of a binomial experiment is
calculated by plugging the value of a into the binomial formula as shown below :
P( x  a)  n Ca p a (1  p) n  a
Continuous Random variable
For a continued random variable the probability of observing one particular value is zero.
i.e. P( x  a )  0
Continuous Probability Distribution
We can only compute probability over an interval of values. Since P( x  a )  0 and P ( x  b)  0 is a
continuous variable,
P ( a  x  b)  P ( a  x  b)
To find probabilities for continuous random variables, we use probability density functions.
Two common types of continuous random variable probability distribution :
1) Uniform distribution.
2) Normal distribution.
A2. Uniform distribution
1
ba
a
b
Note : The area under a probability density function is 1.
Area of rectangle  Height  Width
1  Height  (b  a)
Height 
1
(b  a)
for a uniform distribution
Example 1 : A continuous random variable x is uniformly distributed with 10  x  50 .
(a) Draw a graph of the uniform density function.
(b) What is P(20  x  30) ?
(c) What is P( x  15) ?
Objective B : Normal distribution – Bell-shaped Curve
Example 1: Graph of a normal curve is given.
Use the graph to identify the value of  and  .

  2
  2
  1   1
X
330 430 530 630 730
Example 2 : The lives of refrigerator are normally distributed with mean   14 years and
standard deviation   25 years.
(a) Draw a normal curve and the parameters labeled.
(b) Shade the region that represents the proportion of refrigerator that lasts for more
than 17 years.
(c) Suppose the area under the normal curve to the right x  17 is 0.1151 . Provide
two interpretations of this result.

Chapter 7.2 Applications of the Normal Distribution
Objective A : Area under the Standard Normal Distribution
The standard normal distribution
– Bell shaped curve
–  =0 and  =1
 1
 0
 3.5
2
1
Negative Z
0
1
2
3.5 Z
Positive Z
The random variable for the standard normal distribution is Z .
Use the 𝑍 table (Table V) to find the area under the standard normal distribution. Each value in
the body of the table is a cumulative area from the left up to a specific Z -score.
Probability is the area under the curve over an interval.
The total area under the normal curve is 1.

0
Z
Z
Under the standard normal distribution,
(a) what is the area to the right   0 ?
(b) What is the area to the left   0 ?
Example 1 : Determine the area under the standard normal curve.
(a) that lies to the left of -1.38.
(b) that lies to the right of 0.56.
(c) that lies in between 1.85 and 2.47.
Objective B : Finding the 𝒁-score for a given probability
Area  0.5
Area  0.5
Area  0.5
Example 1 : Find the 𝑍-score such that the area under the standard normal curve to its left is
0.0418.
Example 2 : Find the 𝑍-score such that the area under the standard normal curve to its right is
0.18.
Example 3 : Find the 𝑍-score such that separates the middle 70%.
Objective C : Probability under a Normal Distribution
Step 1 : Draw a normal curve and shade the desired area.
X 
Step 2 : Convert the values X to Z -scores using Z 
.

Step 3 : Use Table V to find the area to the left of each Z -score found in Step 2.
Step 4: Adjust the area from Step 3 to answer the question if necessary.
Example 1 : Assume that the random variable X is normally distributed with mean   50
and a standard deviation   7 . (Note: this is not standard normal curve because
  0 and   1 .)
(a) P( X  58)
(b) P(45  X  63)
Example 2 : GE manufactures a decorative Crystal Clear 60-watt light bulb that it advertises
will last 1,500 hours. Suppose that the lifetimes of the light bulbs are
approximately normal distributed, with a mean of 1,550 hours and a standard
deviation of 57 hours, what proportion of the light bulbs will last more than 1650
hours?
Objective D: Finding the Value of a Normal Random Variable
Step 1 : Draw a normal curve and shade the desired area.
Step 2 : Finding the corresponding area to the left of the cutoff score if necessary.
Step 3 : Use Table V the find the Z -score that corresponds to the area to the left of the cutoff
score.
X 
Step 4 : Obtain X from Z by the formula Z 
or X    z .

Example 1 : The reading speed of 6th grade students is approximately normal (bell-shaped)
with a mean speed of 125 words per minute and a standard deviation of 24 words
per minute.
(a) What is the reading speed of a 6th grader whose reading speed is at the 90%
percentile?
(b) Determine the reading rates of the middle 95% percentile.
Chapter 7.3 Normality Plot
Recall: A set of raw data is given, how would we know the data has a normal distribution? Use
histogram or stem leaf plot.
Histogram is designed for a large set of data.
For a very small set of data it is not feasible to use histogram to determine whether the data
has a bell-shaped curve or not.
We will use the normal probability plot to determine whether the data were obtained from
a normal distribution or not. If the data were obtained from a normal distribution, the data
distribution shape is guaranteed to be approximately bell-shaped for n is less than 30.
Perfect normal curve. The curve is aligned with the dots.
Almost a normal curve. The dots are within the
boundaries.
Not a normal curve. Data is outside the boundaries.
Example 1 : Determine whether the normal probability plot indicates that the sample data
could have come from a population that is normally distributed.
(a)
(b)