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

3. The Standard Normal Distribution 0123-2-3 N(0, 1) z

4. Areas Under the Standard Normal Curve What is the total area under the 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 below 0? What proportion of values of Z will fall above 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 below +1? What proportion of values will fall above +1? What proportion of values will fall above +1? What proportion of values will fall below –1? What proportion of values will fall below –1? What proportion of values will fall between –1 and +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. It turns out that the area to the left of +1 is 0.8413. 0123-2-3 0.8413 z

7. Areas Under the Standard Normal Curve So, what is the area to the right of +1? So, what is the area to the right of +1? 0123-2-3 0.8413 Area? z

8. Areas Under the Standard Normal Curve So, what is the area to the left of -1? So, what is the area to the left of -1? 0123-2-3 0.8413 Area? z

9. Areas Under the Standard Normal Curve So, what is the area between -1 and 1? So, what is the area between -1 and 1? 0123-2-3 0.8413 Area? 0.8413 z

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

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

12. The Standard Normal Table z.00.01.02… ::::… 0.90.81590.81860.8212… 1.00.84130.84380.8461… 1.10.86430.86650.8686… ::::…

13. The area to the left of 1.00 is 0.8413. The area to the left of 1.00 is 0.8413. That means that 84.13% of the population is below 1.00. That means that 84.13% of the population is below 1.00. 0123-2-3 0.8413

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

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

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

17. Standard Normal Areas Use the TI-83 to find the following. Use the TI-83 to find the following. The area to the left of 1.42. The area to the left of 1.42. The area to the right of 0.87. The area to the right of 0.87. The area between –2.14 and +1.36. 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. 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? 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? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015

20. Other Normal Curves For example, if X is N(30, 5), what is the area to the left of 35? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015

21. Other Normal Curves For example, if X is N(30, 5), what is the area to the left of 35? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015 ?

22. Other Normal Curves For example, if X is N(30, 5), what is the area to the left of 35? For example, if X is N(30, 5), what is the area to the left of 35? 30354045252015 0123-2-3 X Z ?

23. Other Normal Curves To determine the area, we need to find out how many standard deviations 35 is above average. 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. 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. 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. 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). The area to the left of 35 in N(30, 5). 30354045252015 0.8413 0123-2-3 X Z

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

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 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). If X is N( ,  ), then (X –  )/  is N(0, 1).

27. Example Let X be N(30, 5). Let X be N(30, 5). What proportion of values of X are below 38? What proportion of values of X are below 38? Compute z = (38 – 30)/5 = 8/5 = 1.6. Compute z = (38 – 30)/5 = 8/5 = 1.6. Find the area to the left of 1.6 under the standard normal curve. Find the area to the left of 1.6 under the standard normal curve. Answer: 0.9452. Answer: 0.9452. Therefore, 94.52% of the values of X are below 38. 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 3 rd and 4 th parameters of the normalcdf function. Use the same procedure as before, except enter the mean and standard deviation as the 3 rd and 4 th parameters of the normalcdf function. For example, 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. 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. 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 above 120? above 140? What percentage of the population has an IQ between 75 and 125? 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 For a normal distribution, what percentages of the population lie within one standard deviation of the mean? within one standard deviation of the mean? within two standard deviations of the mean? within two standard deviations of the mean? within three standard deviations of the mean? within three standard deviations of the mean? What does this tell us about IQ scores? What does this tell us about IQ scores?

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