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

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.

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.

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

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

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?

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

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 1 2 3

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

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

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

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).

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.

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.

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?

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).

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.

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?

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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

Appendix – Using the Standard Normal Table

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.

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

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

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 1 2 3

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

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

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

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

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

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

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