1. Chapter 9 Roadmap
Where are we going?

2. Chapter 9 Roadmap
Construct and interpret confidence interval for estimating an unknown mean

3. Chapter 9 Roadmap
Construct and interpret confidence interval for estimating an unknown mean
Perform significance test for a single mean

4. Chapter 9 Roadmap
Construct and interpret confidence interval for estimating an unknown mean
Perform significance test for a single mean
Construct and interpret a confidence interval for estimating the difference between two means

5. Chapter 9 Roadmap
Construct and interpret confidence interval for estimating an unknown mean
Perform significance test for a single mean
Construct and interpret a confidence interval for estimating the difference between two means
Perform significance test for difference between two means

6. Chapter 9 Roadmap
Use a confidence interval to estimate the mean of the differences from paired samples

7. Chapter 9 Roadmap
Use a confidence interval to estimate the mean of the differences from paired samples
Perform a significance test for the mean of the differences from paired samples

8. Formulas will change a bit, but basic concepts we learned in Chapter 8 remain the same.
Fundamental question: “What are the reasonably likely outcomes from a random sample?”

9. Confidence Interval for a Mean
Section 9.1
Confidence Interval for a Mean

10. Now looking for plausible values of a population mean, rather than plausible values of a population proportion.
Methodology is same as used in Chapter 8.

11. What is still the key ingredient that makes the theory work?

12. What is still the key ingredient that makes the theory work?
Random sampling or random assignment of treatments

13. Recall
is the ___________ _______

14. Recall
is the population mean
is the __________ _________ ______

15. Recall is the population mean is the population standard deviation
x is the ________ _________

16. Recall is the population mean is the population standard deviation
x is the sample mean
s is the __________ _______ _______

17. Recall is the population mean is the population standard deviation
x is the sample mean
s is the sample standard deviation

18. Neither x nor s is likely to be exactly equal to the population parameters.
This would be true no matter how large the sample.

19. Structure of confidence interval for a mean is similar to that for a proportion:

20. Structure of confidence interval for a mean is similar to that for a proportion:
Statistic (critical value) (standard deviation of statistic)

22. What difficulty do you see with using this formula when we take a sample?
x z*

23. What difficulty do you see with using this formula when we take a sample?
x z*
We seldom know value of in real-life situations.
What can we do then?

24. What difficulty do you see with using this formula?
x z*
We seldom know value of in real-life situations.
What can we do then? Substitute s for

25. s for ? How will substituting s for affect your confidence interval?
x z*

26. x ± z*● How will substituting s for affect your confidence interval?
Some samples give an estimate that’s too small: s < confidence interval ___
______

27. x ± z*● How will substituting s for affect your confidence interval?
Some samples give an estimate that’s too small: s < confidence interval too narrow

28. x ± z*● How will substituting s for affect your confidence interval?
Some samples give an estimate that’s too small: s < σ confidence interval too narrow
Some samples give an estimate that’s too large: s > σ confidence interval __
______

29. x ± z*● How will substituting s for affect your confidence interval?
Some samples give an estimate that’s too small: s < σ confidence interval too narrow
Some samples give an estimate that’s too large: s > σ confidence interval too wide

30. s for ?
On average, the sampling distribution of s has its center very near σ, so, on average, the width of the confidence interval is about right.
However . . .

31. s for ?
However, confidence intervals are judged on their capture rate, not on their average width.

32. Capture Rate
What is meant by capture rate?

33. Capture Rate
Capture rate of a method for producing confidence intervals is the chance that the interval will contain, in our case, the true population mean.

34. s for ?
Approximate sampling distribution of s for 1000 samples of size 4 (left) and size 20 (right) for a normally distributed population with = 107.

35. s for ?
Sampling distribution of s is skewed right so the median is smaller than the mean.
Thus, s tends to be smaller than more often than it is larger.
How does this affect width of CI’s?

36. s for ? Thus, s tends to be smaller than more often than it is larger.
This causes the confidence intervals to be too narrow more than half the time.

37. s for ? Thus, s tends to be smaller than more often than it is larger.
This causes the confidence intervals to be too narrow more than half the time.
How does this affect the capture rate?

38. s for ?
This causes the confidence intervals to be too narrow more than half the time.
So the capture rate will be less than its advertised rate ( < 95% for example).

39. If an interval’s true capture rate is smaller than you want it to be, you can get a larger capture rate by using a _______ interval.

40. If an interval’s true capture rate is smaller than you want it to be, you can get a larger capture rate by using a wider interval.

41. If an interval’s true capture rate is smaller than you want it to be, you can get a larger capture rate by using a wider interval.
x z*

42. To increase the width of the interval, replace z
To increase the width of the interval, replace z* with a larger value called t*.
x z*
x t*

43. t-Distribution
To increase the width of the interval, replace z* with a larger value called t*.
t* is critical value for a t-distribution

44. Sometime, read page 565 for more information on t*.

45. Value of t* depends on only two things:
(1) how many observations you have
(2) the capture rate you want

46. To determine t*, you need to use the degrees of freedom (df).

47. To determine t*, you need to use the degrees of freedom (df).
Degrees of freedom is equal to n – 1.

48. To determine t*, you need to use the degrees of freedom (df).
Degrees of freedom is equal to n – 1.
For a sample size of 8, the df are 8 – 1 = 7

50. Find the correct value of t* to use for each of these situations.
95% confidence interval based on n = 10
b) 99% confidence interval based on n = 49
c) 95% confidence interval based on n = 200

51. n-1

52. Find the correct value of t* to use for each of these situations.
95% confidence interval based on n = 10
t* = 2.262
99% confidence interval based on n = 49
c) 95% confidence interval based on n = 200

53. Find the correct value of t* to use for each of these situations.
95% confidence interval based on n = 10
t* = 2.262
b) 99% confidence interval based on n = 49
If df falls between two of those listed in the column on the left, use the smaller df
c) 95% confidence interval based on n = 200

55. Find the correct value of t* to use for each of these situations.
95% confidence interval based on n = 10
t* = 2.262
b) 99% confidence interval based on n = 49
t* = 2.704
c) 95% confidence interval based on n = 200

57. Find the correct value of t* to use for each of these situations.
95% confidence interval based on n = 10
t* = 2.262
b) 99% confidence interval based on n = 44
t* = 2.704
c) 95% confidence interval based on n = 200 t* = 1.984

58. Confidence Interval for a Mean

59. Constructing Confidence Interval for a Mean
Three major actions you need to take:
(1)
(2)
(3)

60. Constructing Confidence Interval for a Mean
Three major actions you need to take:
(1) Check conditions
(2)
(3)

61. Constructing Confidence Interval for a Mean
Three major actions you need to take:
(1) Check conditions
(2) Do computations
(3)

62. Constructing Confidence Interval for a Mean
Three major actions you need to take:
(1) Check conditions
(2) Do computations
(3) Give interpretation in context

63. Check Conditions
Three conditions:
(1)

64. Check Conditions
Three conditions:
(1) Randomness

65. Check Conditions Three conditions: (1) Randomness
Survey: sample must be randomly selected

66. Check Conditions Three conditions: (1) Randomness
Survey: sample must be randomly selected
Experiment: randomly assigned treatments

67. Check Conditions Three conditions:
(2) Normality: how do we check this?

68. Check Conditions Three conditions: Normality: how do we check this?
Can we use n1p1, etc are all at least 5?

69. Check Conditions Three conditions: Normality: how do we check this?
Can we use n1p1, etc are all at least 5?
No, because we are not working with proportions now.

70. Check Conditions Three conditions:
(2) Normality: you must plot the data to see if it’s reasonable to assume that the data came from a normally distributed population

71. Check Conditions Three conditions:
Normality: you must plot the data to see if it’s reasonable to assume that it came from a normally distributed population
The plot should show a somewhat symmetrical distribution with no outliers

72. How can you plot the data?

73. How can you plot the data?
1) dot plot
2) box plot
3) stem-and-leaf plot

74. Dot Plot

75. Box Plot

76. Stem-and-Leaf Plot

77. Check Conditions Three conditions: Normality: Plot data . . . or
Sample must be large enough that the sampling distribution of the sample mean is approximately normal (more on this in Section 9.3)

78. Check Conditions
Three conditions:
(3) Population size:

79. Check Conditions Three conditions:
(3) Population size: For survey, population size should be at least 10 times as large as the sample size

80. Do Computations
A confidence interval for the population mean, , is given by:
x t*

81. Do Computations 8: TInterval
A confidence interval for the population mean, , is given by:
x t*
Can use calculator: STAT, TESTS
8: TInterval