1. LESSON 10: NORMAL DISTRIBUTION
Outline
Normal distribution
Area under the curve, probability, percentile value
Given z find area
Given percentile value find z
Given x find area
Given percentile value find x

2. NORMAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION
If a random variable X with mean  and standard deviation  is normally distributed, then its probability density function is given by

3. NORMAL DISTRIBUTION THE PROBABILITY DENSITY FUNCTION
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500 20 40 60 80
100 x -
VALUES f ( )
Mean, m
=50
SD, s
=10
Area under
the curve
= 1.00
Area between
the vertical lines = P
(40 X
 60)

4. NORMAL DISTRIBUTION EFFECT OF CHANGING STANDARD DEVIATION
0.0000
0.0100
0.0200
0.0300
0.0400
0.0500 50
100
150
200
x-values
f(x)
SD,=10
SD,=15
SD,=20
Mean,= 100

5. NORMAL DISTRIBUTION EFFECT OF CHANGING MEAN
SD,s=10

6. STANDARD NORMAL DISTRIBUTION

7. STANDARD NORMAL DISTRIBUTION

8. STANDARD NORMAL DISTRIBUTION RELATIONSHIP BETWEEN x AND z
If a random variable X is normally distributed with mean  and standard deviation , then

9. STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES, AREA AND PROBABILITY
Example 1.1: Table D, Appendix A, pp shows the area under the curve from Z=-∞ to some z value. For example, the area from Z=-∞ to Z= =1.34 is So,

10. STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES, AREA AND PROBABILITY
Example 1.2: The area shown on the table can be used to get many other areas. For example, using the fact that the area under the curve is 1.0, the area from Z=1.34 to Z= is = So,

11. STANDARD NORMAL DISTRIBUTION TABLE, z-VALUES, AREA AND PROBABILITY
Example 1.3: The area shown on the table can be used to get area between any two z-values. For example, the area from z1=-1.25 to z2=1.34 is =0.8043, where is the area obtained from Table D for z1= So,

12. STANDARD NORMAL DISTRIBUTION GIVEN z, FIND PROBABILITY
Example 2: Find the following:

13. STANDARD NORMAL DISTRIBUTION GIVEN z, FIND PROBABILITY : EXCEL
Excel function NORMSDIST(z) provides the area under the standard normal distribution curve on the left side of z.
Example: NORMSDIST(1.34) = =
To get the area on the left of Z = 1.34, Φ(1.34), use Excel function NORMSDIST(1.34).
Area =?
z=1.34

14. STANDARD NORMAL DISTRIBUTION AREA AND PERCENTILE
A percentile is the value at or below which the stated percentage of units lie. Therefore, percentile corresponds to an area under the curve. For example, if GMAT scores are normally distributed with the 78th percentile 600, then 78% scores are less than 600 and Then, the area on the left of X = 600 is 0.78.
If the 78th percentile is 600, then the area on the left of X=600 is 0.78.
Area = 0.78
X=600

15. STANDARD NORMAL DISTRIBUTION GIVEN AREA OR PERCENTILE, FIND z
Example 3: If return on investment of a fund has a mean 0 and standard deviation 1, find the returns that corresponds to following percentiles:
1. 96th
2. 30th

16. STANDARD NORMAL DISTRIBUTION GIVEN AREA OR PERCENTILE, FIND z : EXCEL
Excel function NORMSINV(p) provides the value of z corresponding to the 100pth percentile. For example, NORMSINV(0.33)= So, for the standard normal distribution, the 33rd percentile is
To get the z value for which area on the left, Φ(z) = 0.33, use Excel function NORMSINV(0.33).
Area
=0.33
z=?

17. NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY
Example 4.1: A retailer has observed that the monthly demand of an item is normally distributed with a mean of 650 and standard deviation of 50 units. What is the probability that the demand of the item in the next month will not exceed 700 units?
1. Compute z
2. Find area from the Table
3. Find probability

18. NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY
Example 4.2: A retailer has observed that the monthly demand of an item is normally distributed with a mean of 650 and standard deviation of 50 units. What is the probability that the demand of the item in the next month will exceed 600 units?
1. Compute z
2. Find area from the Table
3. Find probability

19. NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY
Example 4.3: A retailer has observed that the monthly demand of an item is normally distributed with a mean of 650 and standard deviation of 50 units. What is the probability that the demand of the item in the next month will be between 600 and 700 units?
1. Compute z1 and z2
2. Find areas from the Table
3. Find probability

20. NORMAL DISTRIBUTION GIVEN x, FIND PROBABILITY : EXCEL
Excel function NORMDIST(x,μ,,TRUE) provides the area under the standard normal distribution curve on the left side of x. For example, NORMDIST(700,650,50,TRUE) =
To get the area on the left of X = 700, when μ=600, σ=50, use Excel function NORMSDIST(700,650,50,TRUE).
μ=600
σ=50
Area=?
X = 700

21. NORMAL DISTRIBUTION GIVEN AREA OR PERCENTILE, FIND x
Example 5: A retailer has observed that the monthly demand of an item is normally distributed with a mean of 650 and standard deviation of 50 units. If the retailer wants to meet demand with probability 0.90, how many units should be ordered for the next month? Assume that there is no units in the inventory.
1. Find z from the table
2. Find x

22. STANDARD NORMAL DISTRIBUTION GIVEN AREA OR PERCENTILE, FIND x : EXCEL
Excel function NORMINV(p,μ,) provides 100pth percentile when mean is μ and standard deviation . So, the function also gives that value of X for which the area on the left side of X is p. For example, NORMINV(0.90,650,50) = 714. So, the 90th percentile is 714.
To get the x value for which area on the left = 0.90, use Excel function NORMSINV(0.90,650,50) if mean is 650 and standard deviation 50.
μ=600
σ=50
Area
= 0.90
X = ?

23. READING AND EXERCISES Lesson 10 Reading: Section 7-4, pp. 216-225
7-35, 7-36, 7-37