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 H1: 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 H0: 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

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 = SD / N N = number of pairs

S =

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

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

S = Difference (D) 2 1 Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 D = 7 D2 =13 N = 4 S =

S = Difference (D) 2 1 Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 D = 7 D2 =13 N = 4 7 S =

S = Difference (D) 2 1 Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 D = 7 D2 =13 N = 4 7 S = 13

S = Difference (D) 2 1 Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 D = 7 D2 =13 N = 4 7 S = 13 4 4 - 1

S = Difference (D) 2 1 Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 D = 7 D2 =13 N = 4 7 S = 13 12.25 4 3

.5 = Difference (D) 2 1 Test 1 w/ Pill (X) Mel 3 Alice 5 Vera 4 Flo 3 Test 2 w/o Pill (Y) 1 3 2 D = 7 D2 =13 N = 4 7 .5 = .75 4 3

Step 4: Calculate t observed tobs = (X - Y) / SD SD = SD / 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 H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words Reject H0, and accept H1 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 Time 1 Child1 18 Child2 11 Child3 19 Child4 6 Child5 10 16 10 17 4 11 12

Hypothesis One-tailed Alternative hypothesis H1: time1 > time2 Null hypothesis H0: time1 < or = time2

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

1.21 = (D) 2 1 -1 Time 1 (X) Child1 18 Child2 11 Child3 19 Child4 6 Test 2 (Y) 16 10 17 4 11 12 D = 8 D2 =18 N = 6 8 1.21 = 18 6 6 - 1

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 H0, and accept H1 If tobs does not fall in the critical region: Fail to reject H0

Step 7: Put answer into words Reject H0, and accept H1 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 H1: Prof > 100 Null hypothesis H0:  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 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. 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?