1. CHAPTER 7: NORMAL DISTRIBUTION
Mr. Mark Anthony Garcia, M.S.
Mathematics Department
De La Salle University
CHAPTER 7: NORMAL DISTRIBUTION

2. Normal Distribution
A normal distribution is a probability distribution that plots all of its values in a symmetrical fashion and most of the results are situated around the probability's mean.
Values are equally likely to plot either above or below the mean. Grouping takes place at values that are close to the mean and then tails off symmetrically away from the mean.

3. Normal Distribution
The normal distribution is represented by a bell-shaped curve called the normal curve and the area under the curve represents the probability of the normal random variable X.

4. Situation: Normal Distribution
Let X be the height of a student from DLSU and X is a normal random variable. Suppose that the mean height of all DLSU students is 𝜇=163𝑐𝑚 with standard deviation 𝜎=20𝑐𝑚.
𝝁−𝝈
143
𝝁+𝝈
183
𝝁−𝟐𝝈
123
𝝁+𝟐𝝈
203
𝝁−𝟑𝝈
103
𝝁+𝟑𝝈
223

5. Situation: Normal Distribution

6. Properties of the Normal Curve
It is a bell-shaped curve.
The mode, which is the point on the horizontal axis where the curve is a maximum, occurs at x = μ. This means that the mean is equal to the mode.
The curve is symmetric about a vertical axis through the mean μ. The mean divides the set of data into two equal parts. This means that the mean is equal to the median.

7. Properties of the Normal Curve
The normal curve approaches the horizontal axis asymptotically as we proceed in either direction away from the mean. (The graph approaches the x-axis but the graph will never intersect the x-axis).
The total area under the curve and above the horizontal axis is equal to 1.

8. Formula: Normal Distribution
The formula for the normal distribution is given by

9. Comparing Normal Curves
Consider the figure below.

10. Comparing Normal Curves
Observe that the blue, red and yellow normal curves have the same mean because they are centered at 𝜇=0 but with different heights because of different frequencies. However, the green normal curve has mean 𝜇=−2.

11. Comparing normal curves
Moreover, all the normal curves shown have different variances which measures the dispersion or spread of the values in the data set. This is illustrated by the width of the curve. It can be seen from the figure, that the yellow normal curve has the largest width and with variance 𝜎 2 =5.

12. Comparing Normal Curves
To avoid having normal distributions with different means and standard deviations, we convert the normal random variable X into the standard normal random variable Z.
The standard normal random variable Z has mean equal to zero (𝜇=0) and standard deviation equal to one (𝜎=1).

13. Standard Normal Distribution
To convert the values of the normal random variable X to the standard normal random variable Z, we use the formula given by 𝑍= 𝑋−𝜇 𝜎 .

14. Example: From X to Z
Suppose that X is the normal random variable with mean 𝜇=550 and standard deviation 𝜎=150. What is the probability that X is greater than 600?
In symbols, we have 𝑃(𝑋>600).
Using the formula 𝑍= 𝑋−𝜇 𝜎 , substitute 𝑋=600, 𝜇=550 and 𝜎=150.

15. Example 1: Normal Distribution
Hupper Corporation produces many types of softdrinks, including Orange Cola. The filling machines are adjusted to pour 12 ounces of soda into each 12-ounce can of Orange cola. However, the actual amount of soda poured into each can is not exactly 12 ounces; it varies from can to can. It has been observed that the net amount of soda in such a can has a normal distribution with a mean of 12 ounces and a standard deviation of ounce.

16. Example 1: Normal Distribution
What is the probability that a randomly selected can of Orange Cola contains to ounces of soda?
Probability for a Range
From X Value
11.97
To X Value
11.99
Z Value for 11.97 -2
Z Value for 11.99
P(X<=11.97)
0.0228
P(X<=11.99)
0.2525
P(11.97<=X<=11.99)
0.2297

17. Example 1: Normal Distribution
The probability that a can of Orange Cola will have between to ounces is
If there are 1000 cans of Orange Cola, how many of the 1000 cans will have between to ounces?
The number of cans is =229.7 or approximately 230 cans.

18. Example 1: Normal Distribution
Suppose we delete the last row of the PhStat output. How do we get 𝑃(11.97≤𝑋≤11.99)?
Probability for a Range
From X Value
11.97
To X Value
11.99
Z Value for 11.97 -2
Z Value for 11.99
P(X<=11.97)
0.0228
P(X<=11.99)
0.2525

19. Example 1: Normal Distribution
To determine the probability, we have
𝑃 𝑋≤11.99 −𝑃 𝑋≤11.97
0.2525−0.0228=0.2297

20. Example 1: Normal Distribution
What percentage of the cola contains at least ounces of soda?
𝑃 𝑋> =0.0478
Probability for X <=
X Value
12.025
Z Value
P(X<=12.025)
Probability for X >
X Value
12.025
Z Value
P(X>12.025)
0.0478

21. Example 1: Normal Distribution
Suppose that the output below is the only given output.
Probability for X <=
X Value
12.025
Z Value
P(X<=12.025)

22. Example 1: Normal Distribution
𝑃 𝑋> =1−𝑃(𝑋≤12.025)
𝑃 𝑋> =1−0.9522=0.0478
The probability that the cola will have at least ounces of soda is

23. Example 2: Normal Distribution
The price of diesel oil over the past 24 months is normally distributed with a mean of 41 pesos per liter and standard deviation of 5 pesos per liter.

24. Example 2: Normal Distribution
What is the probability that the price of diesel is at most 34 pesos per liter?
𝑃 𝑋≤34 =0.0808
Probability for X <=
X Value 34
Z Value
-1.4
P(X<=34)
0.0808

25. Example 2: Normal Distribution
For which amount can we find the highest 10% of the diesel prices?
Find X and Z Given Cum. Pctage.
Cumulative Percentage
10.00%
Z Value
X Value
 34.592
Find X and Z Given Cum. Pctage.
Cumulative Percentage
90.00%
Z Value
1.2816
X Value
47.408

26. Example 2: Normal Distribution
Since the highest 10% of the diesel prices occurs at the rightmost part of the normal curve, we get the area at the left which is 90%.
Thus, we use the table with cumulative percentage equal to 90%.

27. Example 2: Normal Distribution
Getting the X value, we have
This means that that the probability that diesel prices is more than pesos per liter is 10% or 0.10.

28. Example 2: Normal Distribution
Suppose that we delete the last row of the table of cumulative percentage. How do we find the value of X?
Find X and Z Given Cum. Pctage.
Cumulative Percentage
90.00%
Z Value
1.2816
X Value
47.408

29. Example 2: Normal Distribution
Using the formula 𝑍= 𝑋−𝜇 𝜎 , we derive X.
Then 𝑋=𝑍𝜎+𝜇.
𝑋= =47.408

30. Exercises: Normal Distribution
The TV ratings of the show The Big Bang Theory are approximately normally distributed with mean 22.7 and standard deviation 7.4.
Probability for X >
X Value
26.1
Z Value
0.5
P(X>26.1)
0.3085
Probability for X <=
X Value
27.6
Z Value
P(X<=27.6)

31. Exercises: Normal Distribution
What is the probability that for a given day the show The Big Bang Theory will obtain a rating of at least 27.6?
Given 15 episodes of the show The Big Bang Theory, how many episodes would get a rating of less than 26.1?

32. Exercises: Normal Distribution
In the November 1990 issue of Chemical Engineering Progress, a study discussed the percent purity of oxygen from a certain supplier. Assume that the mean was with a standard deviation of Assume that the distribution of percent purity was approximately normal.

33. Exercises: Normal Distribution
What percentage of the purity values would you expect to be between 99.5 and 99.7?
Probability for X <=
X Value
99.5
Z Value
-1.375
P(X<=99.5)
Probability for a Range
From X Value
99.5
To X Value
99.7
Z Value for 99.5
-1.375
Z Value for 99.7
1.125
P(X<=99.5)
0.0846
P(X<=99.7)
0.8697
Probability for X >
X Value
99.7
Z Value
1.125
P(X>99.7)
0.1303

34. Exercises: Normal Distribution
What purity value would you expect to exceed exactly 5% of the population?
Find X and Z Given Cum. Pctage.
Cumulative Percentage
5.00%
Z Value
Find X and Z Given Cum. Pctage.
Cumulative Percentage
95.00%
Z Value