What is a paired samples t test

What is a paired samples t test?

1. Paired Samples t-tests

2. Another important test of differences is the t -test for paired samples. This is also known as a t -test for repeated measures or a t -test for matched samples.

3. Another important test of differences is the t -test for paired samples. This is also known as a t-test for repeated measures or a t-test for matched samples .

4. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people,

5. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people,

6. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people, one month later

7. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people, one month later

8. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people , or from two samples that are matched such that there is a one-to-one correspondence between each subject in group one and its matched pair in group two,

9. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people , or from two samples that are matched such that there is a one-to-one correspondence between each subject in group one and its matched pair in group two, one month later Brown Haired Matched Subjects

10. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people , or from two samples that are matched such that there is a one-to-one correspondence between each subject in group one and its matched pair in group two, o ne month later one month later Brown Haired Matched Subjects Red Haired Matched Subjects

11. Whenever two distributions of a dependent variable are highly correlated, either because they are distributions of pre and post tests from the same people , or from two samples that are matched such that there is a one-to-one correspondence between each subject in group one and its matched pair in group two, the appropriate test of differences is the paired samples t -test.

12. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test,

13. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test,

14. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed;

15. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed;

16. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed;

17. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test)

18. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test)

19. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels

20. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels

21. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels which are repeated or matched,

22. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels which are repeated or matched,

23. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels which are repeated or matched, then we use the pair wise t-test to test for differences between the two samples of the dependent variable .

24. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels which are repeated or matched, then we use the pair wise t-test to test for differences between the two samples of the dependent variable . Single-sample t-test Independent Sample t-test Paired-sample t -test

25. If we have a single dependent variable that exists on an interval or ratio scale such as scores on a test, and is reasonably normally distributed ; and a single independent variable (e.g., when you took the test) which has two levels which are repeated or matched, then we use the pair wise t-test to test for differences between the two samples of the dependent variable . Single-sample t-test Independent Sample t-test Paired-sample t-test

26. First we begin with the null hypothesis:

27. First we begin with the null hypothesis : There are no significant difference in pre and post scores (dependent variable) .

28. First we begin with the null hypothesis : There are no significant difference in pre and post scores (dependent variable) . Or: There are no significant difference between group one and its matched group in terms of the dependent variable .

29. First we begin with the null hypothesis : There are no significant difference in pre and post scores (dependent variable) . Or: There are no significant difference between group one and its matched group in terms of the dependent variable .

30. First we begin with the null hypothesis : There are no significant difference in pre and post scores (dependent variable) . Or: There are no significant difference between group one and its matched group in terms of the dependent variable .

31. First we begin with the null hypothesis : There are no significant difference in pre and post scores (dependent variable) . Or: There are no significant difference between group one and its matched group in terms of the dependent variable . no difference

32. Another way to state the null-hypothesis for a paired-sample t -test is by hypothesizing that the difference between the pre-post test or matched pairs is ZERO.

33. Another way to state the null-hypothesis for a paired-sample t -test is by hypothesizing that the difference between the pre-post test or matched pairs is ZERO. Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard?

34. Another way to state the null-hypothesis for a paired-sample t -test is by hypothesizing that the difference between the pre-post test or matched pairs is ZERO. Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? Name _________ The word of the day is ______ The State bird of Oklahoma is ______ The favorite dessert of Queen Elizabeth is ________ The capital of Siam is ___________ The best movie of 1972 was ___________ The worst spelling bee in the world is _______ What is the capital of Mars? How many people fit in a subway car in Tokyo? The best episode of the Doctor Who reboot is ___________________________ The worst thing you can do with silly putty is ________________ How heavy is a 15 pound bowling ball? What is the best flavor of Mamba fruit chews? The weather in Alaska is __________________ Bugs Bunny is funniest dressed up as a _____________ How much does a taxi cost in London? What is the best flavor Jello Pudding pop? How far is Wichita from Baghdad? What is the most important thing to do on Arbor Day? What goes best with vanilla custard? = −

35. Conceptually, aggregating the exact differences between each pair makes most sense and leads to the correct degrees of freedom, sampling distribution and critical value.  

36. Conceptually, aggregating the exact differences between each pair makes most sense and leads to the correct degrees of freedom, sampling distribution and critical value.   Correct Degrees of Freedom 50 people take a pre- test Same 50 take a post test 50 – 1 = 49 degrees of freedom

37.   Correct Sampling Distribution

38.   Correct Sampling Distribution Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. = −

39. and Correct Critical Value

40. and Correct Critical Value

41. The formula for the paired samples t -test is:

42. The formula for the paired samples t -test is: Σ ( X 1 -X 2 ) SE diff t =

43. The formula for the paired samples t -test is: Σ ( X 1 -X 2 ) SE diff t = Paired t-test value or the number of standard error values that separate these two means

44. The formula for the paired samples t -test is: Pre Post 1 7 2 6 1 8 Σ ( X pre - X post ) SE diff t =

45. The formula for the paired samples t -test is: Pre Post 1 7 2 6 1 8 Σ ( X pre - X post ) SE diff t = Subtract each pretest score from each posttest score and the sum them up

46. The formula for the paired samples t -test is: Pre Post 1 7 2 6 1 8 Σ ( X pre - X post ) SE diff t = Difference 6 4 7

47. The formula for the paired samples t -test is: Pre Post 1 7 2 6 1 8 Σ ( X pre - X post ) SE diff t = Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17

48. The formula for the paired samples t -test is: Pre Post 1 7 2 6 1 8 Σ ( X pre - X post ) SE diff t = Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17 17 SE diff t =

49. The formula for the paired samples t -test is: Σ ( X pre - X post ) SE diff t =

50. Σ ( X pre - X post ) SE diff t = Now let’s calculate the Standard E rror of the difference

51. Σ ( X pre - X post ) SE diff t = Now let’s calculate the Standard E rror of the difference It is this statistic (Standard Error of the Difference) that makes it possible to determine if this difference occurred only by chance or if there is a strong probability that they are actually different!

52. One thing to note : If the Standard Error of the Difference is large (say 170) then the t value will be small.

53. One thing to note : If the Standard Error of the Difference is large (say 170) then the t value will be small . For example:

54. One thing to note : If the Standard Error of the Difference is large (say 170) then the t value will be small . For example: Σ ( X pre - X post ) SE diff t =

55. One thing to note : If the Standard Error of the Difference is large (say 170) then the t value will be small . For example: Σ ( X pre - X post ) SE diff t = 17 170 t =

56. One thing to note : If the Standard Error of the Difference is large (say 170) then the t value will be small . For example: Σ ( X pre - X post ) SE diff t = 17 170 t = .01 t =

57. However, if the Standard Error of the Difference is small (say 1.7) then the t value will be larger.

58. However, if the Standard Error of the Difference is small (say 1.7) then the t value will be larger. For example:

59. However, if the Standard Error of the Difference is small (say 1.7) then the t value will be larger. For example: Σ ( X pre - X post ) SE diff t =

60. However, if the Standard Error of the Difference is small (say 1.7) then the t value will be larger. For example: Σ ( X pre - X post ) SE diff t = 17 1.7 t =

61. However, if the Standard Error of the Difference is small (say 1.7) then the t value will be larger. For example: Σ ( X pre - X post ) SE diff t = 17 1.7 t = 10 t =

62. So what effect will a smaller or larger estimated standard error have on whether a result is statistically significantly different or not?

63. So what effect will a smaller or larger estimated standard error have on whether a result is statistically significantly different or not? Well let’s say that our null hypothesis (H o ) is that there is no statistically significant difference between the pre and post test scores below:

64. So what effect will a smaller or larger estimated standard error have on whether a result is statistically significantly different or not? Well let’s say that our null hypothesis (H o ) is that there is no statistically significant difference between the pre and post test scores below: Students Pre Post 1 1 7 2 2 6 3 1 8 mean: 1.3 6.0

65. Of course, the alternative hypothesis would be that there is no statistically significant difference between the two.

66. Of course, the alternative hypothesis would be that there is no statistically significant difference between the two . With a t value of .01

67. Of course, the alternative hypothesis would be that there is no statistically significant difference between the two . With a t value of .01 Σ ( X pre - X post ) SE diff t = 17 170 t = .01 t =

68. Of course, the alternative hypothesis would be that there is no statistically significant difference between the two . With a t value of . 01, and a sample of 3, and therefore degrees of freedom of 2, we would look up the t critical value at a .05 alpha level (a .05 alpha level means that we are willing to consider an outcome to be significant if it were likely to happen 5 times out of 100 (.05 ), in other words, if it were a very rare occurrence. )

69. So we go to our table of t -Distribution Critical Values to find the critical value that needs to be exceeded by our result in order be considered statistically significantly different.

70. So we go to our table of t -Distribution Critical Values to find the critical value that needs to be exceeded by our result in order be considered statistically significantly different.

71. So we go to our table of t -Distribution Critical Values to find the critical value that needs to be exceeded by our result in order be considered statistically significantly different . So, the critical value we need to exceed in order to reject the null hypothesis is 2.920 .

72. So we go to our table of t -Distribution Critical Values to find the critical value that needs to be exceeded by our result in order be considered statistically significantly different . So, the critical value we need to exceed in order to reject the null hypothesis is 2.920 . However, our t-value is

73. So we go to our table of t -Distribution Critical Values to find the critical value that needs to be exceeded by our result in order be considered statistically significantly different . So, the critical value we need to exceed in order to reject the null hypothesis is 2.920 . However, our t-value is .01 t =

74. So we go to our table of t -Distribution Critical Values to find the critical value that needs to be exceeded by our result in order be considered statistically significantly different . So, the critical value we need to exceed in order to reject the null hypothesis is 2.920 . However, our t-value is .01

75. Our t value of 0.1 does not exceed the t critical of 2.920 , therefore we will fail to reject the null-hypothesis (which essentially means to accept the null-hypothesis).

76. On the other hand, when our t value is 10, under the same conditions,

77. On the other hand, when our t value is 10, under the same conditions, Σ ( X pre - X post ) SE diff t = 17 1.7 t = 10 t =

78. On the other hand, when our t value is 10, under the same conditions, our t value of 10 does exceed the t- critical of 2.920 , therefore we will reject the null-hypothesis (which essentially means to accept the alternative hypothesis ) .

79. So when it comes to inferential statistics (inferring meaning to a larger population from a smaller sample), the size of the standard error determines everything.

80. So when it comes to inferential statistics (inferring meaning to a larger population from a smaller sample), the size of the standard error determines everything. Understanding the standard error theoretically may or may not be important for your learning purposes. If it is not, you may want to click quickly through the next 10 slides. If it is important then consider what follows :

81. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples.

82. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. = −

83. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample

84. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample −

85. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample − this sample

86. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample − this sample =

87. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample − this sample = this sample

88. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample

89. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample −

90. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample − this sample

91. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample this sample = −

92. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample this sample this sample = −

93. If we took 1000 samples of pre- and post-tests and subtracted each pre-test sample from each post-test sample we would have a sampling distribution called the sampling distribution of differences between pre- and post-test samples. Sample Mean Distribution of difference between the pre and post samples test scores Sample Mean Distribution of pre -test scores. Sample Mean Distribution of post- test scores. this sample this sample this sample = − etc …

94. If you calculated the standard deviation of this new subtracted sampling distribution you would have the actual standard error we are looking for for this equation.

95. If you calculated the standard deviation of this new subtracted sampling distribution you would have the actual standard error we are looking for for this equation . Since this is almost impossible to do, the standard error will be estimated.

96. Let’s see how this is done with a very simple data set. Let’s begin with our null-hypothesis: “ Post-test scores are not statistically significantly higher than pre-test scores” .

97. Let’s see how this is done with a very simple data set. Let’s begin with our null-hypothesis: “ Post-test scores are not statistically significantly higher than pre-test scores” . Students Pre Post 1 1 7 2 2 6 3 1 8 mean: 1.3 7.0

98. We will calculate each element of the equation below:

99. We will calculate each element of the equation below:

100. Let’s begin with the sum of the differences between the pre and post tests:

101. Let’s begin with the sum of the differences between the pre and post tests:

102. Let’s begin with the sum of the differences between the pre and post tests: Σ ( X pre - X post ) SE diff t =

103. Let’s begin with the sum of the differences between the pre and post tests: Σ ( X pre - X post ) SE diff t = the same

104. Let’s begin with the sum of the differences between the pre and post tests:

105. Let’s begin with the sum of the differences between the pre and post tests: Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17

106. Let’s begin with the sum of the differences between the pre and post tests: 17

107. Just as a contrast, when the difference is smaller, then the sum of all differences will be smaller as well:

108. Just as a contrast, when the difference is smaller, then the sum of all differences will be smaller as well: Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17

109. Just as a contrast, when the difference is smaller, then the sum of all differences will be smaller as well: Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17

110. Just as a contrast, when the difference is smaller, then the sum of all differences will be smaller as well: Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17 Pre Post 4 7 5 6 5 8

111. Just as a contrast, when the difference is smaller, then the sum of all differences will be smaller as well: Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17 Pre Post 4 7 5 6 5 8 Difference 3 1 3

112. Just as a contrast, when the difference is smaller, then the sum of all differences will be smaller as well: Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17 Pre Post 4 7 5 6 5 8 Difference 3 1 3 Sum of all Differences 3 + 1 + 3 = 7

113. Back to our original equation. The next step is to estimate the standard

114. Back to our original equation. The next step is to estimate the standard 17

115. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . 17 3

116. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . 17 3 3

117. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . 17 3 3

118. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Pre Post 1 7 2 6 1 8 Difference 6 4 7

119. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Pre Post 1 7 2 6 1 8 Difference 6 4 7 Squared Difference 36 16 49

120. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Pre Post 1 7 2 6 1 8 Difference 6 4 7 Squared Difference 36 16 49 Sum of all Differences 36 + 16 + 49 = 101 Σd 2

121. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Let‘s plug in our numbers:

122. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . 17 3 3 101

123. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 17 3 3 101

124. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17

125. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . Pre Post 1 7 2 6 1 8 Difference 6 4 7 Sum of all Differences 6 + 4 + 7 = 17 Squared Sum of all Differences 17 2 = 289

126. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . Let‘s plug in our numbers and then do the calculations:

127. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 17 3 3 101 289

128. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 17 303 3 289

129. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 17 14 2

130. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 17 7

131. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 17 2.646

132. Back to our original equation. The next step is to estimate the standard We begin with “n” or the size of the sample, which in this case is 3 . Then we compute Σd 2 . Then we compute ( Σd ) 2 . 6.425

133. So what does a t value of 6.425 mean? Well, that depends on whether it is larger or smaller than the critical t value.

134. So what does a t value of 6.425 mean? Well, that depends on whether it is larger or smaller than the critical t value . Do you remember how we determine the critical t value?

135. So what does a t value of 6.425 mean? Well, that depends on whether it is larger or smaller than the critical t value . Do you remember how we determine the critical t value ? All we need is

136. So what does a t value of 6.425 mean? Well, that depends on whether it is larger or smaller than the critical t value . Do you remember how we determine the critical t value ? All we need is the degrees of freedom ( sample size (3) minus 1 ) and

137. So what does a t value of 6.425 mean? Well, that depends on whether it is larger or smaller than the critical t value . Do you remember how we determine the critical t value ? All we need is the degrees of freedom ( sample size (3) minus 1 ) and the alpha level we are willing to live with ( in this case . 05. This basically means that we are willing to live with being wrong 5 out of 100 times about our decision . )

138. So with a degrees of freedom of 2 and an alpha level of . 05,

139. So with a degrees of freedom of 2 and an alpha level of . 05,

140. So with a degrees of freedom of 2 and an alpha level of .05, our critical t value is: 2.920

141. So with a degrees of freedom of 2 and an alpha level of .05, our critical t value is: 2.920 Since our t value ( 6.245 ) is greater than our critical t value ( 2.920 ), we will reject the null hypothesis and accept the alternative hypothesis that students’ post-test scores are higher than their pre-test scores.

142. To visualize this on a graph we create a t distribution with degrees of freedom of 2. Since the sample size is so small, this will be a much flatter distribution than a normal distribution.

143. To visualize this on a graph we create a t distribution with degrees of freedom of 2. Since the sample size is so small, this will be a much flatter distribution than a normal distribution . Here is an example of a normal distribution :

144. To visualize this on a graph we create a t distribution with degrees of freedom of 2. Since the sample size is so small, this will be a much flatter distribution than a normal distribution . Here is an example of a normal distribution :

145. To visualize this on a graph we create a t distribution with degrees of freedom of 2. Since the sample size is so small, this will be a much flatter distribution than a normal distribution . Here is an example of a normal distribution: And here is an example of a t-distribution with 2 degrees of freedom :

146. To visualize this on a graph we create a t distribution with degrees of freedom of 2. Since the sample size is so small, this will be a much flatter distribution than a normal distribution . Here is an example of a normal distribution: And here is an example of a t-distribution with 2 degrees of freedom :

147. You may recall that as the degrees of freedom increase the t -distribution begins to approximate the normal distribution.

148. You may recall that as the degrees of freedom increase the t -distribution begins to approximate the normal distribution . Degrees of freedom = 5

149. You may recall that as the degrees of freedom increase the t -distribution begins to approximate the normal distribution . Degrees of freedom = 5

150. You may recall that as the degrees of freedom increase the t -distribution begins to approximate the normal distribution . Degrees of freedom = 5 Degrees of freedom = 10

151. You may recall that as the degrees of freedom increase the t -distribution begins to approximate the normal distribution . Degrees of freedom = 5 Degrees of freedom = 10

152. At about degrees of freedom of 30 the t -distribution looks almost identical to the normal or standard or z-distribution.

153. At about degrees of freedom of 30 the t -distribution looks almost identical to the normal or standard or z-distribution.

154. Let’s go back to our example of a t -distribution with 2 degrees of freedom.

155. Let’s go back to our example of a t -distribution with 2 degrees of freedom.

156. Let’s go back to our example of a t -distribution with 2 degrees of freedom . Here is the location of the critical t value of 2.920 .

157. Let’s go back to our example of a t -distribution with 2 degrees of freedom . Here is the location of the critical t value of 2.920 . 2.920

158. Let’s go back to our example of a t -distribution with 2 degrees of freedom . Here is the location of the critical t value of 2.920 . This means that if there really is no statistical difference between the pre- and post-tests , if we were to hypothetically draw 100 samples 95% of the time those sample would be drawn from this part of the distribution. 2.920

159. Let’s go back to our example of a t -distribution with 2 degrees of freedom . Here is the location of the critical t value of 2.920 . This means that if there really is no statistical difference between the pre- and post-tests , if we were to hypothetically draw 100 samples 95% of the time those sample would be drawn from this part of the distribution. 2.920 95% of samples would be on this side of the distribution

160. Let’s go back to our example of a t -distribution with 2 degrees of freedom . Here is the location of the critical t value of 2.920 . This means that if there really is no statistical difference between the pre- and post-tests , if we were to hypothetically draw 100 samples 95% of the time those sample would be drawn from this part of the distribution. And 5% of the samples would be on this side of the critical t value . 2.920

161. And so we would say to ourselves : If the t value lands to the right of 2.920, then we would hypothesize that they are not part of the same distribution but are actually two separate distributions.

162. And so we would say to ourselves : If the t value lands to the right of 2.920, then we would hypothesize that they are not part of the same distribution but are actually two separate distributions. 2.920 pre-tests post-tests

163. And so we would say to ourselves : If the t value lands to the right of 2.920, then we would hypothesize that they are not part of the same distribution but are actually two separate distributions . Since our calculated t value is 6.245 , we will reject the null hypothesis and accept the alternative hypothesis that they are two different distributions. 2.920 critical t value pre-tests post-tests

164. And so we would say to ourselves : If the t value lands to the right of 2.920, then we would hypothesize that they are not part of the same distribution but are actually two separate distributions . Since our calculated t value is 6.245 , we will reject the null hypothesis and accept the alternative hypothesis that they are two different distributions. 2.920 critical t value pre-tests post-tests 6.245 calculated t value

165. Let’s see an example where the pre- and post-test scores are closer to one another . Will their difference be statistically significantly different?

166. Let’s see an example where the pre- and post-test scores are closer to one another . Will their difference be statistically significantly different? Here is the data set:

167. Let’s see an example where the pre- and post-test scores are closer to one another . Will their difference be statistically significantly different? Here is the data set: Students Pre Post 1 6 7 2 6 6 3 7 8 mean: 6.3 7.0

168. Let’s see an example where the pre- and post-test scores are closer to one another . Will their difference be statistically significantly different? Here is the data set: Here is the null hypothesis : “ Post-test scores are not statistically significantly higher than pre-test scores” . Students Pre Post 1 6 7 2 6 6 3 7 8 mean: 6.3 7.0

169. We will calculate again each element of the equation below:

170. We will calculate again each element of the equation below:

171. Let’s begin with the sum of the differences between the pre and post tests:

172. Let’s begin with the sum of the differences between the pre and post tests: Σ ( X pre - X post ) SE diff t = the same

173. Let’s begin with the sum of the differences between the pre and post tests: Pre Post 6 7 6 6 7 8 Difference 1 1 Sum of all Differences 1 + 0 + 1 = 2

174. Plug in the sum of all differences: 2 Sum of all Differences 1 + 0 + 1 = 2

175. The “n” or the sample size is 3 . 2

176. The “n” or the sample size is 3 . 2 3

177. The “n” or the sample size is 3 . 2 3 3

178. Then we compute Σd 2

179. Then we compute Σd 2 Pre Post 6 7 6 6 7 8 Difference 1 1 Squared Difference 1 1 Sum of all Differences 1 + 0 + 1 = 2 Σd 2

180. Let’s plug in our numbers:

181. Let’s plug in our numbers: 2 3 3 2

182. Then we compute ( Σd ) 2

183. Then we compute ( Σd ) 2 Pre Post 6 7 6 6 7 8 Difference 1 1 Sum of all Differences 1 + 0 + 1 = 2 Squared Sum of all Differences 2 2 = 4

184. Let’s plug in our numbers:

185. Let’s plug in our numbers: 2 3 3 2 4

186. Let’s plug in our numbers and then do the calculations: 2 3 3 2 4

187. Let’s plug in our numbers and then do the calculations: 2 6 3 4

188. Let’s plug in our numbers and then do the calculations: 2 2 2

189. Let’s plug in our numbers and then do the calculations: 2 1

190. Let’s plug in our numbers and then do the calculations: 2 1

191. Let’s plug in our numbers and then do the calculations: 2.0

192. So with a degrees of freedom of 2 and an alpha level of . 05,

193. So with a degrees of freedom of 2 and an alpha level of . 05,

194. So with a degrees of freedom of 2 and an alpha level of .05, our critical t value is: 2.920 .

195. So with a degrees of freedom of 2 and an alpha level of .05, our critical t value is: 2.920 . Since our t value ( 2.0 ) is less than our critical t value ( 2.920 ) we will reject the null hypothesis and accept the alternative hypothesis that students’ post-test scores are higher than their pretest scores .

196. Here is the location of the critical t value of 2.920 :

197. Here is the location of the critical t value of 2.920 : 2.920

198. Here is the location of the critical t value of 2.920 : This means that if there really is no statistical difference between the pre and posttests, if we were to hypothetically draw 100 samples, 95% of the time those samples would be drawn from this part of the distribution . 2.920 95% of samples would be on this side of the distribution

199. Here is the location of the critical t value of 2.920 : This means that if there really is no statistical difference between the pre and posttests, if we were to hypothetically draw 100 samples, 95% of the time those samples would be drawn from this part of the distribution . And 5% of the samples would be on this side of the critical t value . 2.920

200. If the t value lands to the left of 2.920 then we would say to ourselves, 2.920

201. If the t value lands to the left of 2.920 then we would say to ourselves, “that is such a common occurrence that I hypothesize that they are the same distribution and not two separate distributions. ” 2.920 Critical t value 2.0 Calculated t value

202. If the t value lands to the left of 2.920 then we would say to ourselves, “that is such a common occurrence that I hypothesize that they are the same distribution and not two separate distributions. ” 2.920 Critical t value 2.0 Calculated t value pre-tests

203. If the t value lands to the left of 2.920 then we would say to ourselves, “that is such a common occurrence that I hypothesize that they are the same distribution and not two separate distributions. ” 2.920 Critical t value 2.0 Calculated t value pre-tests post-tests

204. If the t value lands to the left of 2.920 then we would say to ourselves, “that is such a common occurrence that I hypothesize that they are the same distribution and not two separate distributions. ” Since our t value is 2.0 , we will fail to reject the null hypothesis. 2.920 Critical t value 2.0 Calculated t value pre-tests post-tests

205. In Summary: The paired sample t -test used in hypothesis testing to determine if to two matched samples ( e.g., pre / posttests or matched in some other way ) are statistically significantly different from one another.

206. End of Presentation