So far We have been doing independent samples

So far. . • We have been doing independent samples designs • The observations in one group were not linked to the observations in the other group

Example Philadelphia 52 53 54 61 55 Newport 77 75 67

Dependent Samples Design • Books calls it a “Paired-Samples Design” • This can happen with: – Natural pairs – Matched pairs – Repeated measures

Natural Pairs The pairing of two subjects occurs naturally (e. g. , twins)

Matched Pairs When people are matched on some variable (e. g. , age)

Repeated Measures The same participant is in both conditions

Paired Samples Design • In this type of design you label one level of the variable X and the other Y • There is a logical reason for paring the X value and the Y value

Paired Samples Design • The logic and testing of this type of design is VERY similar to what you have already done!

Example • You just invented a “magic math pill” that will increase test scores. • On the day of the first test you give the pill to 4 subjects. When these same subjects take the second test they do not get a pill • Did the pill increase their test scores?

Hypothesis One-tailed • Alternative hypothesis – H 1: pill > nopill – In other words, when the subjects got the pill they had higher math scores than when they did not get the pill • Null hypothesis – H 0: pill < or = nopill – In other words, when the subjects got the pill their math scores were lower or equal to the scores they got when they did not take the pill

Results Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 2

Step 2: Calculate the Critical t • N = Number of pairs • df = N - 1 • 4 -1=3 • =. 05 • t critical = 2. 353

Step 3: Draw Critical Region tcrit = 2. 353

Step 4: Calculate t observed tobs = (X - Y) / SD

Step 4: Calculate t observed tobs = (X - Y) / SD

Step 4: Calculate t observed tobs = (X - Y) / SD X = 3. 75 Y = 2. 00

Step 4: Calculate t observed tobs = (X - Y) / SD Standard error of a difference

Step 4: Calculate t observed tobs = (X - Y) / SD SD = S D / N N = number of pairs

S=

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 S= Test 2 w/o Pill (Y) 1 3 2 2

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 S= Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1 D = 7 S= D 2 =13 N=4

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1 D = 7 S= 7 D 2 =13 N=4

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1 D = 7 S= 13 7 D 2 =13 N=4

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1 D = 7 S= 13 4 -1 7 D 2 =13 4 N=4

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1 D = 7 S= 7 12. 25 4 13 3 D 2 =13 N=4

Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Difference Test 2 w/o Pill (Y) (D) 1 2 3 2 2 1 D = 7 . 5 = . 75 3 7 D 2 =13 4 N=4

Step 4: Calculate t observed tobs = (X - Y) / SD SD = S D / N N = number of pairs

Step 4: Calculate t observed tobs = (X - Y) / SD. 25=. 5 / 4 N = number of pairs

Step 4: Calculate t observed 7. 0 = (3. 75 - 2. 00) /. 25

Step 5: See if tobs falls in the critical region tcrit = 2. 353

Step 5: See if tobs falls in the critical region tcrit = 2. 353 tobs = 7. 0

Step 6: Decision • If tobs falls in the critical region: – Reject H 0, and accept H 1 • If tobs does not fall in the critical region: – Fail to reject H 0

Step 7: Put answer into words • Reject H 0, and accept H 1 • When the subjects took the “magic pill” they received statistically ( =. 05) higher math scores than when they did not get the pill

Practice • You just created a new program that is suppose to lower the number of aggressive behaviors a child performs. • You watched 6 children on a playground and recorded their aggressive behaviors. You gave your program to them. You then watched the same children and recorded this aggressive behaviors again.

Practice • Did your program significantly lower ( =. 05) the number of aggressive behaviors a child performed?

Results Child 1 Child 2 Child 3 Child 4 Child 5 Child 6 Time 1 18 11 19 6 10 14 Time 2 16 10 17 4 11 12

Hypothesis One-tailed • Alternative hypothesis – H 1: time 1 > time 2 • Null hypothesis – H 0: time 1 < or = time 2

Step 2: Calculate the Critical t • N = Number of pairs • df = N - 1 • 6 -1=5 • =. 05 • t critical = 2. 015

Step 4: Calculate t observed tobs = (X - Y) / SD

Time 1 (X) Child 1 18 Child 2 11 Child 3 19 Child 4 6 Child 5 10 Child 6 14 (D) 2 1 2 2 -1 2 Test 2 (Y) 16 10 17 4 11 12 D = 8 1. 21 = 18 6 -1 8 D 2 =18 6 N=6

Step 4: Calculate t observed tobs = (X - Y) / SD. 49=1. 21 / 6 N = number of pairs

Step 4: Calculate t observed 2. 73 = (13 - 11. 66) /. 49 X = 13 Y = 11. 66 SD =. 49

Step 5: See if tobs falls in the critical region tcrit = 2. 015 tobs = 2. 73

Step 6: Decision • If tobs falls in the critical region: – Reject H 0, and accept H 1 • If tobs does not fall in the critical region: – Fail to reject H 0

Step 7: Put answer into words • Reject H 0, and accept H 1 • The program significantly ( =. 05) lowered the number of aggressive behaviors a child performed.

• “My teacher is an idiot!” • You wonder if the professors at Villanova are more intelligent than the average person. To examine this you collected data from 4 of your teachers. Determine if Villanova professors really have significantly ( =. 05) higher IQs than the average IQ of the general population ( = 100).

Data

Step 1: Write out Hypotheses • Alternative hypothesis – H 1: Prof > 100 • Null hypothesis – H 0: Prof < or = 100

Step 2: Calculate the Critical t • N=4 • df = 3 • =. 05 • tcrit = 2. 353

Step 3: Draw Critical Region tcrit = 2. 353

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 14. 73=29. 45 / 4

Step 4: Calculate t observed tobs = (X - ) / Sx 2. 44 = (136 - 100) / 14. 73 1. 18=14. 4 / 150

Step 5: See if tobs falls in the critical region tcrit = 2. 353

Step 5: See if tobs falls in the critical region tcrit = 2. 353 tobs = 2. 44

Step 6: Decision • If tobs falls in the critical region: – Reject H 0, and accept H 1 • If tobs does not fall in the critical region: – Fail to reject H 0

Step 7: Put answer into words • We reject H 0 and accept H 1. • Professors at Villanova have significantly ( =. 05) higher IQs than the average IQ of the general population ( = 100).

Practice • 10. 15 • Did the type of signal effect response time?