No class on Wednesday 11/1 No class on Friday 11/3 Remember No class on Wednesday 11/1 No class on Friday 11/3

Example You just created a “Smart Pill” and you gave it to 150 subjects. Below are the results you found. Do people who take your “Smart Pill” have significantly ( = .05) greater IQ scores than the average IQ population ( = 100)? X = 103 s = 14.4

Step 1: Write out Hypotheses Alternative hypothesis H1: sample > 100 Null hypothesis H0: sample < or = 100

Step 2: Calculate the Critical t N = 150 df = 149  = .05 tcrit = 1.645

Step 3: Draw Critical Region tcrit = 1.645

Step 4: Calculate t observed tobs = (X - ) / Sx

Step 4: Calculate t observed tobs = (X - ) / Sx Sx = S / N

Step 4: Calculate t observed tobs = (X - ) / Sx 1.18=14.4 / 150

Step 4: Calculate t observed tobs = (X - ) / Sx 2.54 = (103 - 100) / 1.18 1.18=14.4 / 150

Step 5: See if tobs falls in the critical region tcrit = 1.645

Step 5: See if tobs falls in the critical region tcrit = 1.645 tobs = 2.54

Step 6: Decision If tobs falls in the critical region: Reject H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words We reject H0 and accept H1. The average IQ of the people who took your “Smart Pill” is statistically greater ( = .05) than the average IQ of the population.

Practice You create a program that reduces aggressiveness of children. After completing your program you give 15 children the “Aggressive Test” which has an average score of  = 15.6 . Below are the results you found. Did your program significantly ( = .05) reduce the aggressiveness of children? X = 14.5 s = 2.77

Step 1: Write out Hypotheses Alternative hypothesis H1: sample < 15.6 Null hypothesis H0: sample > or = 15.6

Step 2: Calculate the Critical t N = 15 df = 14  = .05 tcrit = -1.761

Step 3: Draw Critical Region tcrit = -1.761

Step 4: Calculate t observed tobs = (X - ) / Sx

Step 4: Calculate t observed tobs = (X - ) / Sx -1.53 = (14.5 – 15.6) / .72 .72 = 2.77/ 15

Step 5: See if tobs falls in the critical region tcrit = -1.761

Step 5: See if tobs falls in the critical region tcrit = -1.761 tobs = -1.53

Step 6: Decision If tobs falls in the critical region: Reject H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words We fail to reject H0 The average aggressiveness score of the children who took your program is not statistically less then ( = .05) the average aggressiveness of the population.

So far. . . We have been doing hypothesis testing with a single sample We find the mean of a sample and determine if it is statistically different than the mean of a population

Basic logic of research \

Start with two equivalent groups of subjects

Treat them alike except for one thing

See if both groups are different at the end

Notice This means that we need to see if two samples are statistically different from each other We can use the same logic we learned earlier with single sample hypothesis testing

Example You just invented a “magic math pill” that will increase test scores. You give the pill to 4 subjects and another 4 subjects get no pill You then examine their final exam grades

Hypothesis Two-tailed Alternative hypothesis H1: pill = nopill In other words, the means of the two groups will be significantly different Null hypothesis H0: pill = nopill In other words, the means of the two groups will not be significantly different

Hypothesis One-tailed Alternative hypothesis H1: pill > nopill In other words, the pill group will score higher than the no pill group Null hypothesis H0: pill < or = nopill In other words, the pill group will be lower or equal to the no pill group

For current example, lets just see if there is a difference Alternative hypothesis H1: pill = nopill In other words, the means of the two groups will be significantly different Null hypothesis H0: pill = nopill In other words, the means of the two groups will not be significantly different

Results Pill Group 5 3 4 No Pill Group 1 2 4 3

Remember before. . . Step 2: Calculate the Critical t df = N -1

Now Step 2: Calculate the Critical t df = N1 + N2 - 2 df = 4 + 4 - 2 = 6  = .05 t critical = 2.447

Step 3: Draw Critical Region tcrit = -2.447 tcrit = 2.447

Remember before. . . Step 4: Calculate t observed tobs = (X - ) / Sx

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2 X1 = 3.75 X2 = 2.50

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2

Standard Error of a Difference Sx1 - x2 When the N of both samples are equal If N1 = N2: Sx1 - x2 = Sx12 + Sx22

Results Pill Group 5 3 4 No Pill Group 1 2 4 3

Standard Deviation S = -1

Standard Deviation Pill Group 5 3 4 No Pill Group 1 2 4 3 X2= 10

Standard Deviation Pill Group 5 3 4 No Pill Group 1 2 4 3 X2= 10

Standard Deviation Pill Group 5 3 4 No Pill Group 1 2 4 3 X2= 10 Sx= .48 Sx= . 645

Standard Error of a Difference Sx1 - x2 When the N of both samples are equal If N1 = N2: Sx1 - x2 = Sx12 + Sx22

Standard Error of a Difference Sx1 - x2 When the N of both samples are equal If N1 = N2: Sx1 - x2 = (.48)2 + (.645)2

Standard Error of a Difference Sx1 - x2 When the N of both samples are equal If N1 = N2: Sx1 - x2 = (.48)2 + (.645)2 = .80

Standard Error of a Difference Raw Score Formula When the N of both samples are equal If N1 = N2: Sx1 - x2 =

X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 Sx1 - x2 =

X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 Sx1 - x2 = 10 15

Sx1 - x2 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15

Sx1 - x2 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15 4 (4 - 1)

Sx1 - x2 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15 56.25 30 25 4 4 12

.80 = X1= 15 X12= 59 N1 = 4 X2= 10 X22= 30 N2 = 4 10 15 59 56.25 30 25 7.75 4 4 12

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2 Sx1 - x2 = .80 X1 = 3.75 X2 = 2.50

Now Step 4: Calculate t observed Sx1 - x2 = .80 X1 = 3.75 X2 = 2.50

Now Step 4: Calculate t observed 1.56 = (3.75 - 2.50) / .80 Sx1 - x2 = .80 X1 = 3.75 X2 = 2.50

Step 5: See if tobs falls in the critical region tcrit = -2.447 tcrit = 2.447

Step 5: See if tobs falls in the critical region tcrit = -2.447 tcrit = 2.447 tobs = 1.56

Step 6: Decision If tobs falls in the critical region: Reject H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words We fail to reject H0. The final exam grades of the “pill group” were not statistically different ( = .05) than the final exam grades of the “no pill” group.

Practice You wonder if people like Pepsi better than Coke ( = .05) You give Pepsi to 5 people and Coke to 5 people. You ask them to rate on a 1 to 5 scale how much they liked their soda.

Results Pepsi 4 5 3 Coke 4 3 2

Hypotheses Alternative hypothesis Null hypothesis H1: Pepsi > Coke Null hypothesis H0: Pepsi = or < Coke

Step 2: Calculate the Critical t df = N1 + N2 - 2 df = 5 + 5 - 2 = 8  = .05 One-tailed t critical = 1.860

Step 3: Draw Critical Region tcrit = 1.860

Now Step 4: Calculate t observed tobs = (X1 - X2) / Sx1 - x2

Sx1 - x2 = X1= 21 X12= 91 N1 = 5 X1 = 4.2 X2= 15 X22= 49 5 (5 - 1)

.58 = X1= 21 X12= 91 N1 = 5 X1 = 4.2 X2= 15 X22= 49 N2 = 5 .58 = 15 21 91 49 5 5 5 (5 - 1)

Step 4: Calculate t observed 2.07 = (4.2 - 3) / .58 Sx1 - x2 = .58 X1 = 4.2 X2 = 3

Step 5: See if tobs falls in the critical region tcrit = 1. 860 tobs = 2.07

Step 6: Decision If tobs falls in the critical region: Reject H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words We Reject H0, and accept H1 People like Pepsi significantly ( = .05) better than Coke.