1. THE NORMAL DISTRIBUTION
CHAPTER 6
THE
NORMAL DISTRIBUTION

2. Preview
Introduction
As stated in Chapter 5, a random variable can be either a discrete or a continuous random variable. We also learned how to handle probability distribution for discrete random variable.
In this chapter, we are going to learn about probability distribution for continuous random variable.
Definition:
Once again, continuous random variables are defined as variables that can assume any values within one or more intervals.
These values are not countable.
These variables can assume infinite number of values because there are infinite number of values in any given interval.

3. Continuous Probability Distribution
Now suppose we randomly selected 2,000 RCC students and collected their heights.
Table 1 is the frequency and relative frequency distribution of student heights.
The height of each student is a continuous random variable. So let x = continuous random variable for the height of students
Table 1: Frequency and Relative Frequency Distribution of Student Heights
Height of a Student (inches), x f RF
60 to less than 62
150
.0750
62 to less than 64
320
.1600
64 to less than 66
480
.2400
66 to less than 68
600
.3000
68 to less than 70
250
.1250
70 to less than 72
200
.1000
=1.0000

4. Continuous Probability Distribution
Table 1: Frequency and Relative Frequency Distribution of Student Heights
Note:
In Chapter 4, we learned that if an experiment is performed n times and an Event A occurred f times, then using the relative frequency concept of probability, the probability that Event A will occur the next time is
Therefore, we can use the relative frequencies in Table 1 as the probability of each class.
The probabilities are not exact because we are using relative frequencies which are based on sample data.
Height of a Student (inches), x f RF
60 to less than 62
150
.0750
62 to less than 64
320
.1600
64 to less than 66
480
.2400
66 to less than 68
600
.3000
68 to less than 70
250
.1250
70 to less than 72
200
.1000
=1.0000

5. Continuous Probability Distribution
The histogram and polygon of the distribution in Table 1 is shown to the right.
As stated in Chapter 2, as the class width gets smaller and the number of classes increases, the polygon becomes a smooth curve.
The smooth curve is an approximation of the probability distribution curve of a continuous random variable.

6. Continuous Probability Distribution
A probability distribution of a continuous random variable must satisfy the following two conditions:
The probability that x assumes a value in any of the intervals (class) ranges from 0 to 1. In other words, the area under a probability distribution curve of a continuous random variable between any two points is between 0 and 1.
The total probability of all intervals within which x can assume a value is That is, the total area under the probability distribution curve of a continuous random variable is always 1.0.

7. Continuous Probability Distribution
General Statement:
The probability that x assumes a value within a certain interval is equal to the area under the curve between the endpoints of the interval. For example, let
So, the probability that the height of a randomly selected student lies between 68 to 70 is given by the area under the curve from x = 68 to x = 70. 68 70

8. Continuous Probability Distribution
General Statement:
The probability of a continuous random variable is always calculated for an interval. Therefore, the probability that x assumes a single value is zero. P(x=a) = 0 P(x=b) = 0 Area of a line is zero as shown in the figure to the right.
Since P(x=a) = 0 and P(x=b) = 0, then,
In other words, the probability that x assumes a value in an interval from a to b is the same whether or not a and b are included.

9. 6.2 THE NORMAL DISTRIBUTION
Definition:
A normal probability distribution has a mean, , and a standard deviation, , and is characterized by its symmetric bell-shaped curve that satisfy the following conditions.
The total area under the curve is 1.0.
The curve is symmetric about the mean. That is, the mean line divides the shaded region into two equal parts; 50% to the left and 50% to the right of the mean.

10. The Normal distribution
The two tails of the curve extend indefinitely in both directions. The curve never touches but get very close to the horizontal line such that the area under the region beyond on the left and on the right is considered to be zero.
From the three graphs, we need only both the mean, and standard deviation, , to find the area under a normal distribution curve.
Each set of result in a family of normal distribution curves as shown to the right.

11. THE STANDARD NORMAL DISTRIBTUION
Definition
A standard normal probability is a normal distribution of a continuous random variable having a mean of zero and a standard deviation of 1. These types of random variables are denoted by z and called z values or z scores.
Note
Although the values of z are negative to the left of the mean, the area under the curve is positive.
A value of z indicates the distance between the mean and the point represented by z in terms of standard deviation. For example, a point with a z score of 2 is to the right of the mean. A point with a z score of -3 is to the left of the mean.
Table A-2 of Appendix A lists the area under the curve to the left of z values from to 3.49.

12. The Standard Normal Distribution
To read the area under the curve from Table A-2 of Appendix A, we perform the following steps:
Divide the z value into two portions.
The first portion consists of:
1st digit to the left decimal point,
The decimal point followed by
The 1st digit to the right of the decimal point
The second portion is made-up of:
Decimal point, followed by a zero &
The 2nd digit after the decimal point.

13. The Standard Normal Distribution
For example, given z = 1.85, then
The 1st portion is 1.8 and
The second portion is .05.
To find the required area, locate the 1st portion under the column for z, and the 2nd portion in the row of z. Then the entry where the row for the 1st portion intersect the column for the 2nd portion is the required area to the left of the z value. 13

14. The Standard Normal Distribution
Example #1
Example #1 – Solution
For the standard normal distribution, find the area within one standard deviation of the mean-that is, the area between

15. The Standard Normal Distribution
Example #2
Find the area under the standard normal curve
a) from z=0 to z= b) between z=0 and z= c) from z=.84 to z=1.95
d) between z=-.57 and z= e) between z=-2.15 and z=1.87
Example #2 – Solution

16. The Standard Normal Distribution
Example #2 – Solution

17. 6.3 APPLICATIONS OF THE NORMAL DISTRIBUTION
Standardizing a Normal Distribution
Introduction
Not all normal random variables possess standard normal distribution. That is, there are normal random variables with mean and standard deviation other than 0 and 1, respectively.
However, we can still use Table A-2 of Appendix A to find the area under a normal distribution curve, provided we convert the normal distribution to standard normal distribution.
In other words, we have to convert the normal random variable, x, to its equivalent z value or z score.
Formula
The formula for converting a normal random variable to a z score is,

18. Standardizing a Normal Distribution
Note
The z value should be rounded up to two decimal places.
From the given formula for z:
z value = 0 if x =
z value is negative if x is to the left of
z value is positive if x is to the right of
The z value for the mean of a normal distribution is always 0.
The z value for an x-value indicates the distance between the mean and the x value in terms of the standard deviation. x
x=a
Normal distribution z
z=0
z=b
Standard distribution

19. Standardizing a Normal Distribution
Example #5
Determine the z value for each of the following x values for a normal distribution with
Example #5 – Solution

20. Standardizing a Normal Distribution
Example #5 – Solution

21. Standardizing a Normal Distribution
Example #6
Find the following areas under a normal distribution curve with
Example #6 – Solution

22. Standardizing a Normal Distribution
Example #6 – Solution

23. Standardizing a Normal Distribution
Example #7
Let x be a continuous random variable that is normally distributed with a mean of 80 and a standard deviation of 12. Find the probability that x assumes a value
Example #7 – Solution

24. Standardizing a Normal Distribution
Example #7 – Solution 24

25. APPLICATIONS OF THE NORMAL DISTRIBUTION
This section deals with how to use what we have learned to solving real problems.
Example #8
Tommy W., a minor league baseball pitcher, is notorious for taking an excessive amount of time between pitches. In fact, his times between pitches are normally distributed with a mean of 36 seconds and a standard deviation of 2.5 seconds. What percentage of his times between pitches are
Example #8 – Solution

26. Application of the Normal Distribution
Example #9
The Bank of Connecticut issues Visa and MasterCard credit cards. It is estimated that the balance on all Visa credit cards issued by the bank have a mean of $845 and a standard deviation of $270. Assume that the balances on all these Visa cards follows a normal distribution
Example #9 – Solution