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

4. Example
Philadelphia 52 53 54 61 55
Newport 77 75 67

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

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

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

8. Repeated Measures
The same participant is in both conditions

9. 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

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

11. 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?

12. 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

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

14. Step 2: Calculate the Critical t
N = Number of pairs
df = N - 1
4 - 1 = 3
 = .05
t critical = 2.353

15. Step 3: Draw Critical Region
tcrit = 2.353

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

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

18. Step 4: Calculate t observed
tobs = (X - Y) / SD
X = 3.75
Y = 2.00

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

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

21. S =

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

23. 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 =

24. 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 =

25. 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 =

26. 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

27. 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

28. 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

29. .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

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

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

32. Step 4: Calculate t observed
7.0 = ( ) / .25

33. Step 5: See if tobs falls in the critical region
tcrit = 2.353

34. Step 5: See if tobs falls in the critical region
tcrit = 2.353
tobs = 7.0

35. 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

36. 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

38. 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.

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

40. Results Time 1 Child1 18 Child2 11 Child3 19 Child4 6 Child5 1016 10 17 4 11 12

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

42. Step 2: Calculate the Critical t
N = Number of pairs
df = N - 1
6 - 1 = 5
 = .05
t critical = 2.015

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

44. 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

45. Step 4: Calculate t observed
tobs = (X - Y) / SD
.49=1.21 / 6
N = number of pairs

46. Step 4: Calculate t observed
2.73 = ( ) / .49
X = 13
Y = 11.66
SD = .49

47. Step 5: See if tobs falls in the critical region
tcrit = 2.015
tobs = 2.73

48. 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

49. Step 7: Put answer into words
Reject H0, and accept H1
The program significantly ( = .05) lowered the number of aggressive behaviors a child performed.

50. “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).

51. Data

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

53. Step 2: Calculate the Critical t
N = 4
df = 3
 = .05
tcrit = 2.353

54. Step 3: Draw Critical Region
tcrit = 2.353

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

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

57. Step 4: Calculate t observed
tobs = (X - ) / Sx
14.73=29.45 / 4

58. Step 4: Calculate t observed
tobs = (X - ) / Sx
2.44 = ( ) / 14.73
1.18=14.4 / 150

59. Step 5: See if tobs falls in the critical region
tcrit = 2.353

60. Step 5: See if tobs falls in the critical region
tcrit = 2.353
tobs = 2.44

61. 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

62. 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).

63. Practice
10.15
Did the type of signal effect response time?