1. Section 9.5
Day 2

2. Wed and Thurs
1st period: meet in Room D174
Wed
5th period meet in Room D174

3. Key Question ?

4. Do we really have two independent samples
Key Question
Do we really have two independent samples or

5. Do we really have two independent samples
Key Question
Do we really have two independent samples or
do we have only one sample of paired data?

6. Components of Test
What are the components of a significance test of the mean difference based on paired observations?

7. Components
What are the components of a significance test of the mean difference based on paired observations?
1) Name test and check conditions

8. Components
What are the components of a significance test of the mean difference based on paired observations?
1) Name test and check conditions
2) State hypotheses

9. Components
What are the components of a significance test of the mean difference based on paired observations?
1) Name test and check conditions
2) State hypotheses
3) Compute test statistic, find P-value, and draw sketch

10. Components
What are the components of a significance test of the mean difference based on paired observations?
1) Name test and check conditions
2) State hypotheses
3) Compute test statistic, find P-value, and draw sketch
4) Write conclusion linked to computations and in context of problem

11. Name Test Name: One-sided significance test of a mean
difference based on paired observations or
Two-sided significance test of a mean difference based on paired observations

12. Check Conditions
Same conditions as for a confidence interval

13. Check Conditions
1) Randomness: For survey, must have a random sample from one population where a “unit” may consist of a pair of twins or same person’s two feet for example.
For experiment, treatments randomly assigned within each pair. If same subject assigned both treatments, treatments must be assigned in random order.

14. Check Conditions
2) Normality: The differences, not the two original samples, must look like it’s reasonable to assume they came from normally distributed population or sample size must be large enough that sampling distribution of sample mean difference is approximately normal.
15/40 guideline can be applied to the differences.

15. Check Conditions
3) Sample size: For survey, size of the population of differences should be at least ten times as large as the sample size.
Condition does not apply to experiment.

16. State Hypotheses

17. State Hypotheses
Null hypothesis almost always is that in the entire population, the mean of the differences is 0.

18. State Hypotheses
Null hypothesis almost always is that in the entire population, the mean of the differences is 0.
In symbols, Ho: = 0

19. State Hypotheses Alternative hypothesis is one of these: Ha: 0

20. Compute test statistic, find P-value, and draw sketch
P-value is based on n - 1 degrees of freedom, where n is the number of differences.

21. Compute test statistic, find P-value, and draw sketch
Use T-Test.

22. Write conclusion linked to computations and in context of problem
Reject null hypothesis if P-value is less than the significance level,

23. Write conclusion linked to computations and in context of problem
Reject null hypothesis if P-value is less than the significance level,
Do not reject null hypothesis if P-value is greater than or equal to the significance level,

24. Write conclusion linked to computations and in context of problem
Reject null hypothesis if P-value is less than the significance level,
Do not reject null hypothesis if P-value is greater than or equal to the significance level,
Write a conclusion that relates to specific situation

25. Page 652, P33

26. Page 652, P33 The repeated measures design shows the
strongest relationship, as it should,
because measurements are made on the
same person.

27. Page 652, P33 The repeated measures design shows the
strongest relationship, as it should,
because measurements are made on the
same person.
The scatterplot for the completely
randomized design shows the weakest
relationship, or very little association
between the measurements.

28. Page 652, P33(b)
One-sided significance test for the difference of two means

29. Page 652, P33(b)
Conditions:
Two treatments were randomly assigned to the available experimental units.

30. Page 652, P33(b)
Two treatments were randomly assigned to the available experimental units.
Boxplots fairly symmetric with no outliers so reasonable to assume both samples came from normally distributed populations.
Note: if not given in problem, then you must show your graphs

31. Page 652, P33(b)
Two treatments were randomly assigned to the available experimental units.
Boxplots fairly symmetric with no outliers so reasonable to assume data came from normally distributed populations.
Experiment so only two conditions to check.

32. Ho:

33. Ho: µstand = µsit , where µstand is the true mean pulse rate for standing and µsit is the true mean pulse rate for sitting for this group of subjects Ha:

34. Ho: µstand = µsit , where µstand is the true mean pulse rate for standing and µsit is the true mean pulse rate for sitting for this group of subjects Ha: µstand > µsit standing increases mean pulse rate

35. Completely randomized design
Page 652, P33(b)
Completely randomized design
2-SampTTest
Inpt: Data Stats
x1: 75.71
sx1: Pooled: No Yes
n1: Calculate
x2: 64.57
sx2: 9.33
n2: 14

36. Completely randomized design
Page 652, P33(b)
Completely randomized design
t ; P-value

37. Page 652, P33(b) I reject the null hypothesis because the
P-value of is less than the significance
level of 0.05.

38. Page 652, P33(b) I reject the null hypothesis because the
P-value of is less than the significance
level of 0.05.
If all the students could have been given each of the standing and sitting treatments, there is sufficient evidence to support the claim that standing increases the mean pulse rate compared to sitting.

39. Matched Pairs Design (MPD)
Page 652, P33(c)
Matched Pairs Design (MPD)

40. Name Test
Name: One-sided significance test of a mean difference based on paired observations

41. Page 652, P33(c) Matched Pairs Design Check conditions:
Treatments randomly assigned within each matched pair.

42. Page 652, P33(c) Matched Pairs Design Check conditions:
Treatments randomly assigned within each matched pair.
The boxplot of the differences shows skewness, but not enough to be a serious problem for a sample this large. There are no outliers. So it’s reasonable to assume the differences came from a normally distributed population

43. Page 652, P33(c)
Matched Pairs Design Ho: µd = 0, where µd is the mean difference in pulse rates if each treatment could have been assigned to each subject within each pair

44. Page 652, P33(c)
Matched Pairs Design Ho: µd = 0, where µd is the mean difference in pulse rates if each treatment could have been assigned to each subject within each pair Ha: µd > 0

45. Page 652, P33(c) Matched Pairs Design T-Test Inpt: Data Stats : 0
List: L3
Freq: 1
Calculate

46. Page 652, P33(c)
Matched Pairs Design
t ; P-value

47. Page 652, P33(c) Matched Pairs Design
I reject the null hypothesis because the
P-value of is less than the
significance level of 0.05.

48. Page 652, P33(c) If each treatment could have been assigned
Matched Pairs Design
I reject the null hypothesis because the
P-value of is less than the
significance level of 0.05.
If each treatment could have been assigned
to each subject in each pair, there is
sufficient evidence to support the claim
that standing increases the mean pulse rate
compared to sitting.

49. Repeated Measures Design (RMD)
Page 652, P33 (d)
Repeated Measures Design (RMD)

50. Name Test
Name: One-sided significance test of a mean difference based on paired observations

51. Repeated Measures Design (RMD)
Page 652, P33 (d)
Repeated Measures Design (RMD)
Check conditions:
Both treatments randomly assigned to each subject

52. Repeated Measures Design (RMD)
Page 652, P33 (d)
Repeated Measures Design (RMD)
Check conditions:
Both treatments randomly assigned to each subject
The boxplot of the differences shows little skewness so it’s reasonable to assume the differences came from a normally distributed population.

53. Page 652, P33 (d)
Repeated Measures Design (RMD) Ho: µd = 0, where µd is the mean difference in pulse rates if each treatment could have been given in both orders to each subject Ha: µd > 0

54. Repeated Measures Design
Page 652, P33(d)
Repeated Measures Design
T-Test
Inpt: Data Stats
: 0
List: L3
Freq: 1
Calculate

55. Repeated Measures Design (RMD)
Page 652, P33 (d)
Repeated Measures Design (RMD)
t ; P-value = X
The test statistic is quite large and would be
difficult to see in a sketch.

56. Repeated Measures Design (RMD)
Page 652, P33 (d)
Repeated Measures Design (RMD)
I reject the null hypothesis because the
P-value of X is less than the
significance level of 0.05.

57. Repeated Measures Design (RMD)
Page 652, P33 (d)
Repeated Measures Design (RMD)
I reject the null hypothesis because the
P-value of X is less than the
significance level of 0.05.
There is sufficient evidence to support
the claim that standing increases the
mean pulse rate compared to sitting.

58. Page 652, P33 (e) In this study, all three designs produced
statistically significant results, with decreasing
P-values of 0.005, , and less than
So, the evidence in favor of the alternative
hypothesis increased as you reduced variation
between pairs by first matching on base pulse
rate and then using repeated measures. This is
fairly typical of the performance of these three
designs.

59. Page 653, P34 (a)

60. Page 653, P34 (a)
There is no reason to pair a particular bird with a particular fish, so these are independent samples.

61. Page 655, E70

62. Page 655, E70 a. You do not have two independent samples;
because the measurements of the distance are for
the fifth launch and tenth launch for each team,
this is a repeated measures design. You should
use a one-sample test of the mean of
the differences.

63. Page 655, E70 a. You do not have two independent samples;
because the measurements of the distance are for
the fifth launch and tenth launch for each team,
this is a repeated measures design. You should
use a one-sample test of the mean of
the differences.
A one-sided test should be used because you
want to test whether the teams improved, which
would mean the distance on the tenth launch
would be longer.

64. Page 655, E70 b. Check conditions: This is not a random sample
of teams and the paired observations are not in
random order; this is an observational study of
teams.

65. Page 655, E70 b. Check conditions: This is not a random sample
of teams and the paired observations are not in
random order; this is an observational study of
teams. The boxplot of the differences is basically
symmetric so can assume differences came from normally distributed population.

66. Page 655, E70 b. Check conditions:
We do not know if there were 60 or more teams that launched gummy bears.

67. Page 655, E70 b. The conditions have not been met for
a significance test for paired differences but
the test result can still be useful.

68. Page 655, E70 State your hypotheses:
H0 : μd = 0, where μd is the mean of the differences between the distance on the tenth launch and the distance on the fifth launch
Ha :

69. Page 655, E70 State your hypotheses:
H0 : μd = 0, where μd is the mean of the differences between the distance on the tenth launch and the distance on the fifth launch
Ha : μd > 0

70. Page 655, E70

71. Page 655, E70 Write your conclusion in context, linked to
your computations: I do not reject the null
hypothesis because the P-value of
is greater than = 0.01.
There is not statistically significant evidence
that the teams improved from launch 5 to
launch 10.

72. Page 655, E70
c. What type of error might we have made?

73. Page 655, E70

74. Page 655, E70 d. The result may not match your intuition.
You would still expect some improvement
with practice, although you would expect less
improvement from the fifth launch to the tenth than
from the first launch to the tenth.
At some point you might expect that more practice
is not going to cause any more noticeable
improvement, but you might not expect that to
occur already after the fifth launch.

75. Questions?