Understanding the Standard Normal Distribution

1. The Normal Distribution Lecture 20 Section 6.3.1 Mon, Oct 9, 2006

2. The Standard Normal Distribution • The standard normal distribution – The normal distribution with mean 0 and standard deviation 1. • It is denoted by the letter Z. • Therefore, Z is N(0, 1).

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

4. Areas Under the Standard Normal Curve • 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?

5. Areas Under the Standard Normal Curve • What proportion of values will fall below +1? • What proportion of values will fall above +1? • What proportion of values will fall below –1? • What proportion of values will fall between –1 and +1?

6. Areas Under the Standard Normal Curve • It turns out that the area to the left of +1 is 0.8413. 0.8413 z -3 -2 -1 0 1 2 3

7. Areas Under the Standard Normal Curve • So, what is the area to the right of +1? Area? 0.8413 z -3 -2 -1 0 1 2 3

8. Areas Under the Standard Normal Curve • So, what is the area to the left of -1? Area? 0.8413 z -3 -2 -1 0 1 2 3

9. Areas Under the Standard Normal Curve • So, what is the area between -1 and 1? Area? 0.8413 0.8413 z -3 -2 -1 0 1 2 3

10. Areas Under the Standard Normal Curve • We will use two methods. • Standard normal table. • The TI-83 function normalcdf.

11. The Standard Normal Table • See pages 406 – 407 or pages A-4 and A-5 in Appendix A. • The entries in the table are the areas to the left of the z-value. • To find the area to the left of +1, locate 1.00 in the table and read the entry.

12. The Standard Normal Table

13. The Standard Normal Table • The area to the left of 1.00 is 0.8413. • That means that 84.13% of the population is below 1.00. 0.8413 -3 -2 -1 0 1 2 3

14. a a b a 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.

15. Standard Normal Areas • Use the Standard Normal Tables to find the following. • The area to the left of 1.42. • The area to the right of 0.87. • The area between –2.14 and +1.36.

16. 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.

17. Standard Normal Areas • Use the TI-83 to find the following. • The area to the left of 1.42. • The area to the right of 0.87. • The area between –2.14 and +1.36.

18. Other Normal Curves • The standard normal table and the TI-83 function normalcdf are for the standard normal distribution. • If we are working with a different normal distribution, say N(30, 5), then how can we find areas under the curve?

19. Other Normal Curves • For example, if X is N(30, 5), what is the area to the left of 35? 15 20 25 30 35 40 45

20. Other Normal Curves • For example, if X is N(30, 5), what is the area to the left of 35? 15 20 25 30 35 40 45

21. Other Normal Curves • For example, if X is N(30, 5), what is the area to the left of 35? ? 15 20 25 30 35 40 45

22. Other Normal Curves • For example, 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 0 1 2 3

23. Other Normal Curves • To determine the area, we need to find out how many standard deviations 35 is above average. • Since  = 30 and  = 5, we find that 35 is 1 standard deviation above average. • Thus, we may look up 1.00 in the standard normal table and get the correct area. • The number 1.00 is called the z-score of 35.

24. Other Normal Curves • The area to the left of 35 in N(30, 5). 0.8413 X 15 20 25 30 35 40 45 Z -3 -2 -1 0 1 2 3

25. Z-Scores • Z-score, or standard score, of an observation – The number of standard deviations from the mean to the observed value. • Compute the z-score of x as or • Equivalently or

26. Areas Under Other Normal Curves • If a variable X has a normal distribution, then the z-scores of X have a standard normal distribution. • If X is N(, ), then (X – )/ is N(0, 1).

27. 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: 0.9452. • Therefore, 94.52% of the values of X are below 38.

28. TI-83 – Areas Under Other 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. • For example, normalcdf(-E99, 38, 30, 5) = 0.9452.

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

30. The “68-95-99.7 Rule” • For a normal distribution, what percentages of the population lie • within one standard deviation of the mean? • within two standard deviations of the mean? • within three standard deviations of the mean? • What does this tell us about IQ scores?

31. The Empirical Rule • The well-known Empirical Rule is similar, but more general. • If X has a “mound-shaped” distribution, 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.