1. Welcome to Week 08 College Statistics

2. http://www. andreabalt

3. Now, for something even more profound…

4. TRUTH

5. Question on a true/false test:
Truth
Question on a true/false test:
US presidents are named
“Barack” T ___ F ___

6. Question on a true/false test:
Truth
Question on a true/false test:
US presidents are named
“George” T ___ F ___

7. Question on a true/false test:
Truth
Question on a true/false test:
US presidents are male
T ___ F ___

8. Question on a true/false test:
Truth
Question on a true/false test:
US presidents are at least
35 years old T ___ F ___

9. So, it is easier to be “false” than to be “true”
Truth
So, it is easier to be “false” than to be “true”

10. Truth
So, it is easier to be “false” than to be “true” To be “true” a statement must be true in all cases

11. Truth
So, it is easier to be “false” than to be “true” To be “true” a statement must be true in all cases If not, it is “false”

12. We live in a world where the “truth” is not always known
The Bad News…
We live in a world where the “truth” is not always known

13. In a court of law, you never REALLY know if someone is guilty or not

14. Even with an eyewitness, the witness could be:
Guilt
Even with an eyewitness, the witness could be:
Mistaken
Lying

15. There is always a level of UNCERTAINTY
Guilt
There is always a level of UNCERTAINTY

16. Two standards of proof of guilt in US courts of law:
Criminal cases – “beyond a reasonable doubt”
Civil cases – “a preponderance of the evidence”

17. In probability terms, legal authorities estimate:
Guilt
In probability terms, legal authorities estimate:
“beyond a reasonable doubt” is 98-99% likelihood of guilt based on the evidence
(Thanks to Ronald B. Standler)

18. (Thanks to Ronald B. Standler)
Guilt
In probability terms, legal authorities estimate:
“a preponderance
of the evidence” is
just a hair over
50% likelihood of
guilt
(Thanks to Ronald B. Standler)

19. There is always the possibility of being WRONG
Guilt
There is always the possibility of being WRONG

20. Guilt
In the US, a suspect is considered “innocent until proven guilty in a court of law”

21. Guilt
BUT… if a suspect is not found guilty in court, are they called: - innocent ? - not guilty ?

22. Guilt
The suspect is called “not guilty” because the defense hasn’t proved their innocence… it is just that the prosecution was unable to prove their guilt!

23. Questions?

24. Hypothesis Tests hypothesis
In science, an “educated guess” is called a:
hypothesis

25. Hypothesis Tests hypothesis test
In science, using experimental evidence to see if it supports a hypothesis is called a:
hypothesis test

26. Hypothesis Tests Vikings in Newfoundland?

27. Hypothesis Tests
Our hypothesis was that we wouldn’t find anything…

28. Hypothesis Tests We rejected that hypothesis!

29. Hypothesis Tests
In practice, we often do hypothesis tests “undercover” as

30. Hypothesis Tests
In practice, we often do hypothesis tests “undercover” as CONFIDENCE INTERVALS

31. Hypothesis Tests
Suppose we had a 95% confidence interval: 5 ≤ µ ≤ 10 Suppose our hypothesis was that µ = 7 Is 7 a likely value for µ given our confidence interval?

32. Hypothesis Tests
Because µ = 7 is in our confidence interval: 5 ≤ µ ≤ 10 It is a possible value given our data

33. Hypothesis Tests
Because µ = 7 is in our confidence interval: 5 ≤ µ ≤ 10 It is a possible value given our data… but so are µ = 6 µ = 8 µ = 9.3 µ = 5.1 µ = …

34. Hypothesis Tests
What if the hypothesized value for µ was 11? 5 ≤ µ ≤ 10 We are 95% confident that µ cannot be 11 given our evidence

35. Hypothesis Tests
We reject the hypothesis that µ = 11 if 5 ≤ µ ≤ 10 with 95% confidence

36. Hypothesis Tests
We reject the hypothesis that µ = 11 if 5 ≤ µ ≤ 10 with 95% confidence We will be wrong to do this 5% of the time (100% - 95%)

37. Hypothesis Tests
We reject the hypothesis that µ = 11 if 5 ≤ µ ≤ 10 with 95% confidence We will be wrong to do this 5% of the time (100% - 95%) The amount of time we are willing to be wrong is called our “α-level”

38. Hypothesis Tests
The confidence interval can be used to test hypothesized values of µ using the mean, standard deviation and sample size of our sample data

39. Hypothesis Tests
Whether we can reject a hypothesis or not depends on how variable our data are!
Not too different… Very different!

40. (see why variability is important?)
Hypothesis Tests
(see why variability is important?)

41. Questions?

42. Hypothesis Tests
Rejecting a hypothesis is a strong statement We have evidence to show µ ≠ 11

43. Hypothesis Tests
If the value is included in the confidence interval, you cannot make a strong statement We haven’t proved µ = 7 (because it could be a wide range of numbers within the interval)

44. Hypothesis Tests
So, we merely “fail to reject” the hypothesis

45. Hypothesis Tests
Our exercise on human temperature last week was a test of the hypothesis that normal human temperature is 98.6°

46. Hypothesis Tests
98.6°

47. Hypothesis Tests
PROJECT QUESTION
You have a hypothesis that normal human body temperature is 98.6° You have experimentally found that measured using an IR thermometer, the inside mouth temperature is between 89.9° and 93.6° with 95% confidence

48. Hypothesis Tests
PROJECT QUESTION
89.9° < temp < 93.6° What do you decide about your hypothesis that human body temperature is 98.6°?

49. Hypothesis Tests
PROJECT QUESTION
89.9° < temp < 93.6° Reject your hypothesis that human body temperature is 98.6° What is the probability that you are wrong to reject this hypothesis?

50. Hypothesis Tests
PROJECT QUESTION
89.9° < temp < 93.6° Reject your hypothesis that human body temperature is 98.6° What is the probability that you are wrong to reject this hypothesis? 5%

51. Questions?

52. Hypothesis Tests
You need to answer an "Is there a difference" question Is there any difference between these two populations? Does some new process improve results?

53. Hypothesis Tests
There is a TRUE (population) answer to your question

54. Hypothesis Tests
You will NEVER find the true answer to most questions because of variability: in your measurements in the data itself in the measuring tool in the samples you get from your population

55. Hypothesis Tests
Are the statistics demons mad at you today?

56. Hypothesis Tests
Reality of life: things aren't clear, certain and constant They are fuzzy, uncertain and variable

57. Hypothesis Tests
This is the basis of statistics - getting a measurement of the fuzziness - "variability"

58. Hypothesis Tests
A hypothesis is a statement about the properties of the population

59. Hypothesis Tests
It may be obtained from theory, hearsay, historical studies, etc.

60. Hypothesis Tests
A null hypothesis states "there is no difference between populations" or "a process has no effect"

61. Hypothesis Tests
It is symbolized: H0

62. Hypothesis Tests
Because it is easier to prove something false than to prove it true… H0 is the hypothesis we want to reject

63. Hypothesis Tests
We want to show the populations are different or the process has an effect - called the alternate hypothesis or Ha

64. Hypothesis Tests
Usually we set Ha before H0, since it is the one we are interested in

65. Hypothesis Tests
Null hypotheses about population means are typically like: μ = some value

66. (called one-tailed tests)
Hypothesis Tests
Alternative hypotheses about means can be:
μ ≠ some value
(called a two-tailed test) μ < some value
μ > some value
(called one-tailed tests)

67. Hypothesis Tests
A two-tailed test will reject H0 either if the experimental values we get are too high or too low

68. Hypothesis Tests
α is split between the upper and lower tails

69. Hypothesis Tests
A one-tailed test will reject H0 only on the side we think is likely to be true

70. Hypothesis Tests
You will be able to reject H0 more often for a one-tailed test – if you pick the right tail!

71. Hypothesis Tests
PROJECT QUESTION
Your owner's manual says you should be getting 30 mpg highway After owning the car for six months, you are only getting 27 mpg highway

72. Hypothesis Tests
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? What is H0? What is Ha?

73. Hypothesis Tests
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is H0? What is Ha?

74. Hypothesis Tests
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is H0? μ = 30 mpg What is Ha?

75. Hypothesis Tests
PROJECT QUESTION
Is that different enough to reject the company's claim? What is your α-level? 5% or 0.05 What is H0? μ = 30 mpg What is Ha? μ < 30 mpg

76. We could also write it as: H0: μ ≥ 30 mpg Ha: μ < 30 mpg
Hypothesis Tests
PROJECT QUESTION
We could also write it as: H0: μ ≥ 30 mpg Ha: μ < 30 mpg

77. Is this a one-tailed or a two- tailed test?
Hypothesis Tests
PROJECT QUESTION
Is this a one-tailed or a two- tailed test?

78. Hypothesis Tests
PROJECT QUESTION
Is this a one-tailed or a two- tailed test? one-tailed Is it right-tailed or left-tailed?

79. Is it right-tailed or left-tailed? left-tailed
Hypothesis Tests
PROJECT QUESTION
Is it right-tailed or left-tailed? left-tailed

80. Questions?

81. Hypothesis Tests
The experiment is designed to gather valid information to test the likelihood of that null hypothesis being true

82. Hypothesis Tests
So, since we want to show the null hypothesis is NOT true, we want to show that getting the results we got (if the null hypothesis IS true) is very unlikely

83. If we get those “unlikely” data
Hypothesis Tests
If we get those “unlikely” data

84. Hypothesis Tests
Then we reject the null hypothesis and have statistically proved our alternative hypothesis and

85. Hypothesis Tests
CELEBRATE!

86. Hypothesis Tests But any experiment runs the risk of weird results
The objective of hypothesis testing is to estimate the likelihood of weird results

87. Hypothesis Tests
One type of error consists of rejecting a true hypothesis We call this a “Type 1 error”

88. Hypothesis Tests
If this happens, people will accuse us of rigging our data to prove Ha
So, we want
this to happen
very rarely

89. Hypothesis Tests
The probability of a Type 1 error is called

90. Hypothesis Tests
The probability of a Type 1 error is called an α-level

91. Hypothesis Tests
Typically we use α = 0.05 (5%) or 0.01 (1%)

92. Hypothesis Tests
If is crucial to set your α-level before you do the experiment or gather any data

93. Hypothesis Tests
If is crucial to set your α-level before you do the experiment or gather any data Otherwise people will accuse you of setting the level to ensure rejecting H0

94. Hypothesis Tests
You can make the opposite mistake: fail to reject H0 when it is false Called a Type 2 error The probability of this kind of error is denoted by β (beta)

95. Hypothesis Tests
We HATE Type 2 errors because they mean we FAILED to prove what we wanted to prove! (Remember, we want to reject H0)

96. Hypothesis Tests
Usually β is computed after the experiment (not determined in advance by the experimenter)

97. Hypothesis Tests
Generally, the larger α value that you permit, the smaller β value you will end up with Conversely, if you demand a smaller α, you will usually get a larger β

98. Hypothesis Tests
Other factors affecting β: sample size it’s harder to detect a difference if it’s really really tiny

99. Hypothesis Tests
Likelihood of making the right decision and rejecting the (false) null hypothesis is: 1 - β called the “power of the test”

100. Hypothesis Tests
For a given α value, we would like the test to be as "powerful" as possible, give us the best chance of rejecting a false null hypothesis

101. Which is more powerful, a one-tailed or a two-tailed test?
Hypothesis Tests
PROJECT QUESTION
Which is more powerful, a one-tailed or a two-tailed test?

102. Hypothesis Tests
PROJECT QUESTION
Which is more powerful, a one-tailed or a two-tailed test? one-tailed (if you guess the right side)

103. Hypothesis Tests
This setup allows us only to disprove a null hypothesis, never prove it

104. Hypothesis Tests
We either disprove it, or we fail to disprove it

105. Hypothesis Tests
We NEVER accept the null hypothesis

106. Hypothesis Tests
"Fail to reject" the null hypothesis is the default-decision

107. Hypothesis Testing
This results not from evidence in favor of the null hypothesis but from the absence of evidence against it

108. Hypothesis Tests
Rejecting the null hypothesis is a strong conclusion, stating that (with no more than α given chance of error) the null hypothesis is wrong

109. Hypothesis Tests
The confidence interval for the hypothesis test will be kinda the opposite of what we did before Now we will create a confidence interval for 𝒙 based on our hypothesized value for μ and see if our 𝒙 falls in it

110. Hypothesis Tests
How to do it!

111. Hypothesis Tests
How to do it! Set your α-level (how often you are willing to be wrong)

112. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0

113. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data (for a confidence interval, you need the hypothesized μ, s and n (or se)

114. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value (for two-sided α=5% it is ≈2)

115. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value Calculate the confidence interval using μ rather than 𝒙

116. Hypothesis Tests
How to do it! Set your α-level Define your Ha and H0 Get your data Find your critical value Calculate the confidence interval for 𝒙 The test will be: Is 𝒙 in it?

117. Hypothesis Tests
PROJECT QUESTION
Back to our mpg! H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 And suppose we know that: se = 4 mpg

118. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg Are we going to reject H0 for values of x greater than 30 or less than 30?

119. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg If the critical value for a one-sided confidence interval test at the 5% level is 1.64, create a test of our hypothesis

120. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg Reject H0 if x < 30 - (1.64)(4) < What is our conclusion?

121. Hypothesis Tests
PROJECT QUESTION
H0: μ ≥ 30 mpg Ha: μ < 30 mpg x = 27 se = 4 mpg Reject H0 if x < 30 - (1.64)(4) < What is our conclusion? fail to reject H0

122. Questions?

123. Hypothesis Tests
If you reject H0 with an α-level of 0.05, we also say our x value is “significant at the .05 level” or we say we found a “significant difference”

124. Hypothesis Tests
We can make our x more likely to be significant by (as usual): TAKING A LARGER SAMPLE SIZE

125. Hypothesis Tests
Because we can “cheat the system” by taking a huge sample size that will find any teeny, tiny difference to be significant, we have a backup plan

126. Hypothesis Tests
We also set levels of “practical significance” - what numerical difference would convincingly show a significant difference

127. Hypothesis Tests
These levels of practical significance come from our knowledge of the variables we are measuring

128. Hypothesis Tests
If we had taken a sample of 10,000,000 to calculate our mpg average and se, we could easily have had an se of 0.1 mpg Probably we wouldn’t really think that was a significant difference in mileage

129. Hypothesis Tests
A practically significant difference would be the amount in mpg that you would think is different enough from 30 mpg to be important

130. Hypothesis Tests
We set a level of practical significance at the same time we set the α-level

131. Hypothesis Tests
PROJECT QUESTION
What would be your level of practically significant difference for mpg?

132. Questions?