Practice A research was interested in the relation between stress and humor. Below are data from 8 subjects who completed tests of these two traits. Are these two variables related to each other? How much stress would a person probably experience if they had no sense of humor (i.e., score = 0)? How about if they had a high level of humor (i.e., score = 15)? Stress                Sense of Humor                         4                             2                       10                             8                       12                           11                         5                             3                         7                             8                         6                             7                         2                             3                       14                           13

Practice r = .91 Y = .77 + .98(Humor) .77 = .77 + .98(0) 15.47 = .77 + .98(15) You don’t want to have a sense of humor

What is the probability of picking an ace?

Probability =

What is the probability of picking an ace? 4 / 52 = .077 or 7.7 chances in 100

Every card has the same probability of being picked

What is the probability of getting a 10, J, Q, or K?

(.077) + (.077) + (.077) + (.077) = .308 16 / 52 = .308

What is the probability of getting a 2 and then after replacing the card getting a 3 ?

(.077) * (.077) = .0059

What is the probability that the two cards you draw will be a black jack?

10 Card = (.077) + (.077) + (.077) + (.077) = .308 Ace after one card is removed = 4/51 = .078 (.308)*(.078) = .024

Practice What is the probability of rolling a “1” using a six sided dice? What is the probability of rolling either a “1” or a “2” with a six sided dice? What is the probability of rolling two “1’s” using two six sided dice?

Practice What is the probability of rolling a “1” using a six sided dice? 1 / 6 = .166 What is the probability of rolling either a “1” or a “2” with a six sided dice? What is the probability of rolling two “1’s” using two six sided dice?

Practice What is the probability of rolling a “1” using a six sided dice? 1 / 6 = .166 What is the probability of rolling either a “1” or a “2” with a six sided dice? (.166) + (.166) = .332 What is the probability of rolling two “1’s” using two six sided dice?

Practice What is the probability of rolling a “1” using a six sided dice? 1 / 6 = .166 What is the probability of rolling either a “1” or a “2” with a six sided dice? (.166) + (.166) = .332 What is the probability of rolling two “1’s” using two six sided dice? (.166)(.166) = .028

Next step Is it possible to apply probabilities to a normal distribution?

Theoretical Normal Curve -3 -2 -1  1 2  3 

Theoretical Normal Curve -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

We can use the theoretical normal distribution to determine the probability of an event. For example, do you know the probability of getting a Z score of 0 or less? .50 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

We can use the theoretical normal distribution to determine the probability of an event. For example, you know the probability of getting a Z score of 0 or less. .50 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

With the theoretical normal distribution we know the probabilities associated with every z score! The probability of getting a score between a 0 and a 1 is .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

What is the probability of getting a score of 1 or higher? .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

These values are given in Table C on page 400 .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

To use this table look for the Z score in column A Column B is the area between that score and the mean Column B .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

To use this table look for the Z score in column A Column C is the area beyond the Z score .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

The curve is symmetrical -- so the answer for a positive Z score is the same for a negative Z score Column B Column C .3413 .3413 .1587 .1587 -3 -2 -1  1 2  3  Z-scores -3 -2 -1 0 1 2 3

Practice What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? Beyond z = 2.25? Between the mean and z = -1.45

Practice What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? .2123 Beyond z = 2.25? Between the mean and z = -1.45

Practice What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? .2123 Beyond z = 2.25? .0122 Between the mean and z = -1.45

Practice What proportion of the normal distribution is found in the following areas (hint: draw out the answer)? Between mean and z = .56? .2123 Beyond z = 2.25? .0122 Between the mean and z = -1.45 .4265

Practice What proportion of this class would have received an A on the last test if I gave A’s to anyone with a z score of 1.25 or higher? .1056

Note This is using a hypothetical distribution Due to chance, empirical distributions are not always identical to theoretical distributions If you sampled an infinite number of times they would be equal! The theoretical curve represents the “best estimate” of how the events would actually occur

Theoretical Distribution

Empirical Distribution based on 52 draws

Empirical Distribution based on 52 draws

Empirical Distribution based on 52 draws

Theoretical Normal Curve 

Empirical Distribution

Empirical Distribution

Empirical Distribution

PROGRAM http://www.mathsisfun.com/data/quincunx.html

Theoretical Normal Curve  Normality frequently occurs in many situations of psychology, and other sciences

Practice #7.7 #7.8 #7.9

Practice 7.7 7.8 7.9 .0668 Normal distribution A = .0832; B = .2912; C = .4778 7.9 Empirical

Theoretical Normal Curve  Normality frequently occurs in many situations of psychology, and other sciences

Putting it together Remember that many empirical distributions are approximately normal

Putting it together Thus you can compute z scores from raw scores and use the theoretical normal distribution (Table C) to estimate the probability of that score!

Remember Remember how to convert raw scores to Z scores

Z-score Z scores have a mean of 0 Z scores have a standard deviation of 1

Example: IQ Mean IQ = 100 Standard deviation = 15 What proportion of people have an IQ of 120 or higher?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 120 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (120 - 100) / 15 = 1.33 120 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = 1.33; Column C = .0918 120 .0918 -3 -2 -1  1 2  3 

Example: IQ A proportion of .0918 or 9.18 percent of the population have an IQ above 120. What proportion of the population have an IQ below 80?

Step 1: Sketch out question -3 -2 -1  1 2  3 

Step 1: Sketch out question 80 -3 -2 -1  1 2  3 

Step 2: Calculate Z score (80 - 100) / 15 = -1.33 80 -3 -2 -1  1 2  3 

Step 3: Look up Z score in Table Z = -1.33; Column C = .0918 80 .0918 -3 -2 -1  1 2  3 

Example: IQ A proportion of .0918 or 9.18 percent of the population have an IQ below 80. In a class with 600 children how many probably have an IQ below 80?

Example: IQ A proportion of .0918 or 9.18 percent of the population have an IQ below 80. In a class with 600 children how many probably have an IQ below 80? (.0918) * 600 = 55.08 or 55 children