1. The Normal Distribution
Lecture 20
Section 6.3.1
Wed, Feb 20, 2008

2. The “68-95-99.7 Rule” For any normal distribution,
Approximately 68% of the values lie within one standard deviation of the mean.
Approximately 95% of the values lie within two standard deviations of the mean.
Approximately 99.7% of the values lie within three standard deviations of the mean.

3. The Empirical Rule
The well-known Empirical Rule is similar, but more general.
If a distribution has a “mound shape”, then
Approximately 68% lie within one standard deviation of the mean.
Approximately 95% lie within two standard deviations of the mean.
Nearly all lie within three standard deviations of the mean.

4. The Standard Normal Distribution
It is denoted by the letter Z.
That is, Z is N(0, 1).

5. The Standard Normal Distribution1 2 3 -1 -2 -3
N(0, 1) z

6. Areas Under the Standard Normal Curve
Easy questions:
What is the total area under the curve?
What proportion of values of Z will fall below 0?
What proportion of values of Z will fall above 0?

7. Areas Under the Standard Normal Curve
Harder question:
What proportion of values will fall below +1?

8. Areas Under the Standard Normal Curve
It turns out that the area to the left of +1 is
0.8413 z -3 -2 -1 1 2 3

9. Areas Under the Standard Normal Curve
So, what is the area to the right of +1?
Area?
0.8413 z -3 -2 -1 1 2 3

10. Areas Under the Standard Normal Curve
So, what is the area to the left of -1?
Area? z -3 -2 -1 1 2 3

11. Areas Under the Standard Normal Curve
So, what is the area between -1 and 1?
Area?
0.8413 z -3 -2 -1 1 2 3

12. Areas Under the Standard Normal Curve
There are two methods to finding standard normal areas:
The TI-83 function normalcdf.
Standard normal table.
We will use the TI-83 (unless you want to use the table).

13. TI-83 – Standard Normal Areas
Press 2nd DISTR.
Select normalcdf (Item #2).
Enter the lower and upper bounds of the interval.
If the interval is infinite to the left, enter –E99 as the lower bound.
If the interval is infinite to the right, enter E99 as the upper bound.
Press ENTER.

14. Standard Normal Areas Use the TI-83 to find the following.
The area between -1 and +1.
The area to the left of +1.
The area to the right of +1.

15. Other Normal Curves
If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve?

16. TI-83 – Area Under Normal Curves
Use the same procedure as before, except enter the mean and standard deviation as the 3rd and 4th parameters of the normalcdf function.
Find area between 25 and 38 in the distribution N(30, 5).

17. IQ Scores Intelligence Quotient. Understanding and Interpreting IQ.
IQ scores are standardized to have a mean of 100 and a standard deviation of 15.
Psychologists often assume a normal distribution of IQ scores as well.

18. IQ Scores
What percentage of the population has an IQ above 120? above 140?
What percentage of the population has an IQ between 75 and 125?

19. Bag A vs. Bag B Suppose we have two bags, Bag A and Bag B.
Each bag contains millions of vouchers.
In Bag A, the values of the vouchers have distribution N(50, 10). In Bag B, the values of the vouchers have distribution N(80, 15).

20. Bag A vs. Bag B
H0: Bag A
H1: Bag B 30 40 50 60 70 80 90
100
110

21. Bag A vs. Bag B We select one voucher at random from one bag.
H0: Bag A
H1: Bag B 30 40 50 60 70 80 90
100
110

22. Bag A vs. Bag B
If its value is less than or equal to $65, then we will decide that it was from Bag A.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110

23. Bag A vs. Bag B
If its value is less than or equal to $65, then we will decide that it was from Bag A.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110
Acceptance Region

24. Bag A vs. Bag B
If its value is less than or equal to $65, then we will decide that it was from Bag A.
H0: Bag A
H1: Bag B 30 40 50 60 65 70 80 90
100
110
Acceptance Region
Rejection Region

25. Bag A vs. Bag B What is ? H0: Bag A H1: Bag B 30 40 50 60 65 70 80 90
100
110

26. Bag A vs. Bag B
What is ?
H0: Bag A  30 40 50 60 65 70 80 90
100
110

27. Bag A vs. Bag B What is ? H0: Bag A H1: Bag B 30 40 50 60 65 70 80 90
100
110

28. Bag A vs. Bag B
What is ?
H1: Bag B  30 40 50 60 65 70 80 90
100
110

29. Bag A vs. Bag B
If the distributions are very close together, then  and  will be large.
H0: Bag A
H1: Bag B
N(60, 10)
N(70, 15) 30 40 50 60 65 70 80 90
100
110

30. Bag A vs. Bag B
If the distributions are very similar, then  and  will be large.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

31. Bag A vs. Bag B
If the distributions are very similar, then  and  will be large.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

32. Bag A vs. Bag B
Similarly, if the distributions are far apart, then  and  will both be very small.
H0: Bag A
H1: Bag B
N(45, 10)
N(90, 15) 30 40 50 60 65 70 80 90
100
110

33. Bag A vs. Bag B
Similarly, if the distributions are far apart, then  and  will both be very small.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

34. Bag A vs. Bag B
Similarly, if the distributions are far apart, then  and  will both be very small.
H0: Bag A
H1: Bag B  30 40 50 60 65 70 80 90
100
110

35. Z-Scores Z-score, or standard score Compute the z-score of x as or
Equivalently

36. Areas Under Other Normal Curves
If a variable X has a normal distribution, then the z-scores of X have a standard normal distribution.

37. Example
Let X be N(30, 5).
What proportion of values of X are below 38?
Compute z = (38 – 30)/5 = 8/5 = 1.6.
Find the area to the left of 1.6 under the standard normal curve.
Answer:
Therefore, 94.52% of the values of X are below 38.

38. Appendix – Using the Standard Normal Table

39. The Standard Normal Table
See pages 406 – 407 or pages A-4 and A-5 in Appendix A.
The table is designed for the standard normal distribution.
The entries in the table are the areas to the left of the z-value.

40. The Standard Normal Table
To find the area to the left of +1, locate 1.00 in the table and read the entry. z
.00
.01
.02 … :
0.9
0.8159
0.8186
0.8212
1.0
0.8413
0.8438
0.8461
1.1
0.8643
0.8665
0.8686

41. The Standard Normal Table
To find the area to the left of 2.31, locate 2.31 in the table and read the entry. z
.00
.01
.02 … :
2.2
0.9861
0.9864
0.9868
2.3
0.9893
0.9896
0.9898
2.4
0.9918
0.9920
0.9922

42. The Standard Normal Table
The area to the left of 1.00 is
That means that 84.13% of the population is below 1.00.
0.8413 -3 -2 -1 1 2 3

43. The Three Basic Problems
Find the area to the left of a:
Look up the value for a.
Find the area to the right of a:
Look up the value for a; subtract it from 1.
Find the area between a and b:
Look up the values for a and b; subtract the smaller value from the larger. a b a

44. Standard Normal Areas
Use the Standard Normal Tables to find the following.
The area between and
The area to the left of
The area to the right of

45. Tables – Area Under Normal Curves
If X is N(30, 5), what is the area to the left of 35? 15 20 25 30 35 40 45

46. Tables – Area Under Normal Curves
If X is N(30, 5), what is the area to the left of 35? 15 20 25 30 35 40 45

47. Tables – Area Under Normal Curves
If X is N(30, 5), what is the area to the left of 35? ? 15 20 25 30 35 40 45

48. Tables – Area Under Normal Curves
If X is N(30, 5), what is the area to the left of 35? ? X 15 20 25 30 35 40 45 Z -3 -2 -1 1 2 3

49. Tables – Area Under Normal Curves
If X is N(30, 5), what is the area to the left of 35?
0.8413 X 15 20 25 30 35 40 45 Z -3 -2 -1 1 2 3