1. Last Time Binomial Distribution Political Polls Hypothesis Testing
Excel Computation
Political Polls
Strength of evidence
Hypothesis Testing
Yes – No Questions

2. Administrative Matter
Midterm I, coming Tuesday, Feb. 24
(will say more later)

3. Reading In Textbook Approximate Reading for Today’s Material:
Pages ,
Approximate Reading for Next Class:
Pages , , ,

4. Haircut?
Why?
Website:

5. Haircut?

6. Hypothesis Testing
Example: Suppose surgery cures (a certain type of) cancer 60% of time
Q: is eating apricot pits a more effective cure?

7. (i.e. proportion of people cured)
Hypothesis Testing
E.g. Pits vs. Surgery
Let p be “cure rate” of pits
(i.e. proportion of people cured)

8. Hypothesis Testing E.g. Pits vs. Surgery Let p be “cure rate” of pits
(H0 & H1? New method needs to
“prove it’s worth”
so put burden of proof on it)

9. Hypothesis Testing E.g. Pits vs. Surgery Let p be “cure rate” of pits
H0: p < vs H1: p ≥ 0.6
Recall cure rate of surgery
(competing treatment)

10. Hypothesis Testing E.g. Pits vs. Surgery Let p be “cure rate” of pits
H0: p < vs H1: p ≥ 0.6
(OK to be sure of “at least as good”,
since pits nicer than surgery)

11. Hypothesis Testing
H0: p < vs H1: p ≥ 0.6

12. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were cured by pits

13. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were cured by pits
I.e.: “best guess about p” is:

14. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were cured by pits
I.e.: “best guess about p” is:

15. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were cured by pits
I.e.: “best guess about p” is:
Looks
Better?

16. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were cured by pits
I.e.: “best guess about p” is:
But is it conclusive?

17. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose observe X = 11, out of 15 were cured by pits
I.e.: “best guess about p” is:
But is it conclusive?
Or just due to
sampling variation?

18. Hypothesis Testing
Approach: Define
“p-value” =

19. Hypothesis Testing Approach: Define
“p-value” = “observed significance level”

20. Hypothesis Testing Approach: Define
“p-value” = “observed significance level”
= “significance probability”

21. Hypothesis Testing Approach: Define
“p-value” = “observed significance level”
= “significance probability”
= P[seeing something as
unusual as 11 | H0 is true]

22. Hypothesis Testing “p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is true]

23. Hypothesis Testing “p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is true]
Note: for

24. Hypothesis Testing “p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is true]
Note: for could use “X/n = 0.733”

25. Hypothesis Testing “p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is true]
Note: for could use “X/n = 0.733”,
but this depends too much on n

26. (look at example illustrating this)
Hypothesis Testing
“p-value” = “observed significance level”
= P[seeing something as
unusual as 11 | H0 is true]
Note: for could use “X/n = 0.733”,
but this depends too much on n
(look at example illustrating this)

27. Class Example 4 For X ~ Bi(n,0.6): n P(X/n = 0.6) P(X/n >= 0.6) 5
0.346
0.317 10
0.251
0.367 30
0.147
0.422
100
0.081
0.457
300
0.047
0.475
1000
0.026
0.486
3000
0.015
0.492
10000
0.008
0.496

28. Class Example 4 For X ~ Bi(n,0.6): Computed using Excel: n
P(X/n = 0.6)
P(X/n >= 0.6) 5
0.346
0.317 10
0.251
0.367 30
0.147
0.422
100
0.081
0.457
300
0.047
0.475
1000
0.026
0.486
3000
0.015
0.492
10000
0.008
0.496
For X ~ Bi(n,0.6):
Computed using
Excel:

29. Class Example 4 For X ~ Bi(n,0.6): Note: these go to 0,
even at “most likely
value” n
P(X/n = 0.6)
P(X/n >= 0.6) 5
0.346
0.317 10
0.251
0.367 30
0.147
0.422
100
0.081
0.457
300
0.047
0.475
1000
0.026
0.486
3000
0.015
0.492
10000
0.008
0.496

30. Class Example 4 For X ~ Bi(n,0.6): Note: these go to 0,
even at “most likely
value”
So “small” is
not conclusive n
P(X/n = 0.6)
P(X/n >= 0.6) 5
0.346
0.317 10
0.251
0.367 30
0.147
0.422
100
0.081
0.457
300
0.047
0.475
1000
0.026
0.486
3000
0.015
0.492
10000
0.008
0.496

31. Class Example 4 For X ~ Bi(n,0.6): But for these “small” is conclusive
P(X/n = 0.6)
P(X/n >= 0.6) 5
0.346
0.317 10
0.251
0.367 30
0.147
0.422
100
0.081
0.457
300
0.047
0.475
1000
0.026
0.486
3000
0.015
0.492
10000
0.008
0.496

32. Class Example 4 For X ~ Bi(n,0.6): But for these “small” is conclusive
(so use range,
not value) n
P(X/n = 0.6)
P(X/n >= 0.6) 5
0.346
0.317 10
0.251
0.367 30
0.147
0.422
100
0.081
0.457
300
0.047
0.475
1000
0.026
0.486
3000
0.015
0.492
10000
0.008
0.496

33. Hypothesis Testing “p-value” = “observed significance level”
= P[seeing 11 or more
unusual | H0 is true]

34. Hypothesis Testing “p-value” = “observed significance level”
= P[seeing 11 or more
unusual | H0 is true]
So use:
= P[X ≥ 11 | H0 is true]

35. Hypothesis Testing
“p-value” = P[X ≥ 11 | H0 is true]

36. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true]
What to use here?

37. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true]
What to use here?
Recall: H0: p < 0.6

38. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true]
What to use here?
Recall: H0: p < 0.6
How does P[X ≥ 11 | p] depend on p?

39. Hypothesis Testing
How does P[X ≥ 11 | p] depend on p?

40. Hypothesis Testing How does P[X ≥ 11 | p] depend on p?
Calculated in Class EG 4b: p
P(X >= 11|p)
0.2
0.000
0.3
0.001
0.4
0.009
0.5
0.059
0.6
0.217
0.7
0.515
0.8
0.836

41. Hypothesis Testing How does P[X ≥ 11 | p] depend on p?
Bigger assumed p
goes with
Bigger Probability
i.e. less conclusive p
P(X >= 11|p)
0.2
0.000
0.3
0.001
0.4
0.009
0.5
0.059
0.6
0.217
0.7
0.515
0.8
0.836

42. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]

43. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
So, to be “sure” of conclusion, use largest available value of P[X ≥ 11 | p]

44. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
So, to be “sure” of conclusion, use largest available value of P[X ≥ 11 | p]
Thus, define:
“p-value” = P[X ≥ 11 | p = 0.6]

45. Hypothesis Testing “p-value” = P[X ≥ 11 | H0 is true] =
= P[X ≥ 11 | p < 0.6]
So, to be “sure” of conclusion, use largest available value of P[X ≥ 11 | p]
Thus, define:
“p-value” = P[X ≥ 11 | p = 0.6]
(since “=” gives safest result)

46. Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6]

47. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] Generally: use
= P[seeing something as
unusual as X = 11 | H0 is true]

48. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] Generally: use
= P[seeing something as
unusual as X = 11 | H0 is true]
Here use boundary between H0 & H1

49. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] Generally: use
= P[seeing something as
unusual as X = 11 | H0 is true]
Here use boundary between H0 & H1
(above e.g. p = 0.6)

50. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6]
Now calculate numerical value

51. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6]
Now calculate numerical value
(already done above,
Class EG 4)

52. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
Now calculate numerical value
(already done above,
Class EG 4)

53. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
Now calculate numerical value
(already done above,
Class EG 4)
How to interpret?

54. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error when H0 is rejected

55. (i.e. when conclusion is made)
Hypothesis Testing
“p-value” = P[X ≥ 11 | p = 6] =
Intuition: p-value reflects chance of error when H0 is rejected
(i.e. when conclusion is made)

56. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error when H0 is rejected
(i.e. when conclusion is made)
(based on available evidence)

57. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
Intuition: p-value reflects chance of error when H0 is rejected
(i.e. when conclusion is made)
(based on available evidence)
When p-value is small, it is safe to make a
firm conclusion

58. Hypothesis Testing
For small p-value, safe to make firm conclusion

59. Hypothesis Testing For small p-value, safe to make firm conclusion
How small?

60. Hypothesis Testing For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff

61. Hypothesis Testing For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:

62. Hypothesis Testing For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
Reject H0 when p-value < 0.05

63. (just an agreed upon value,
Hypothesis Testing
For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
Reject H0 when p-value < 0.05
(just an agreed upon value,
but very widely used)

64. Hypothesis Testing For small p-value, safe to make firm conclusion
How small?
Approach 1: Traditional (& legal) cutoff
Called here “Yes-No”:
Reject H0 when p-value < 0.05
(but sometimes want different values,
e.g. your airplane is safe to fly)

65. Hypothesis Testing Approach 1: “Yes-No”
Reject H0 when p-value < 0.05

66. Hypothesis Testing Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
Terminology: say results are “statistically significant”, when this happens

67. Hypothesis Testing Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
Terminology: say results are “statistically significant”, when this happens
Sometimes specify a value α
Greek letter “alpha”

68. Hypothesis Testing Approach 1: “Yes-No”
Reject H0 when p-value < 0.05
Terminology: say results are “statistically significant”, when this happens
Sometimes specify a value α
as the cutoff (different from 0.05)

69. Hypothesis Testing Approach 2: “Gray Level”
Idea: allow “shades of conclusion”

70. Hypothesis Testing Approach 2: “Gray Level”
Idea: allow “shades of conclusion”
e.g. Do p-val = and p-val = represent very different levels of evidence?

71. Hypothesis Testing Approach 2: “Gray Level”
Idea: allow “shades of conclusion”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong evidence

72. Hypothesis Testing Approach 2: “Gray Level”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong evidence

73. Hypothesis Testing Approach 2: “Gray Level”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong evidence
stronger when closer to 0.01

74. Hypothesis Testing Approach 2: “Gray Level”
Use words describing strength of evidence:
0.1 < p-val: no evidence
0.01 < p-val < 0.1 marginal evidence
p-val < 0.01 very strong evidence
stronger when closer to 0.01
weaker when closer to 0.1

75. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217 Bottom Line:
Yes-No: can not reject H0, since
0.217 > 0.05
i.e. no firm evidence pits better than surgery
Gray level: not much indicated

76. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than surgery
Gray level: not much indicated

77. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than surgery
Gray level: not much indicated
Practical Issue: since 73% = observed rate for pits > 60% (surgery),

78. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than surgery
Gray level: not much indicated
Practical Issue: since 73% = observed rate for pits > 60% (surgery), may want to gather more data

79. Hypothesis Testing “p-value” = P[X ≥ 11 | p = 6] = 0.217
No firm evidence pits better than surgery
Gray level: not much indicated
Practical Issue: since 73% = observed rate for pits > 60% (surgery), may want to gather more data, might show value of pits

80. Research Corner
Medical Imaging – Another Fun Example
Cornea Data

81. Research Corner Medical Imaging – Another Fun Example Cornea Data
Cornea = Outer surface
of eye

82. Research Corner Medical Imaging – Another Fun Example Cornea Data
Cornea = Outer surface
of eye
“Curvature” important to
vision

83. Research Corner Medical Imaging – Another Fun Example Cornea Data
Cornea = Outer surface
of eye
“Curvature” important to
vision
Study heat map showing
curvature

84. Research Corner Cornea Data Heat map shows curvature
Each image is one person

85. Research Corner Cornea Data Heat map shows curvature
Each image is one person
Understand “population
variation”?

86. Research Corner Cornea Data Heat map shows curvature
Each image is one person
Understand “population
variation”?
(too messy for brain
to summarize)

87. Research Corner
Cornea Data
Approach: Principal
Component Analysis

88. Research Corner Cornea Data Approach: Principal Component Analysis
Idea: follow “direction” in
image space,

89. Research Corner Cornea Data Approach: Principal Component Analysis
Idea: follow “direction” in
image space, that highlights
population features

90. Research Corner
Cornea Data
Population features

91. Research Corner Cornea Data Population features Overall curvature
(hot – cold)

92. Research Corner Cornea Data Population features Overall curvature
(hot – cold)
With the rule astigmatism
(figure 8 pattern)

93. Research Corner Cornea Data Population features Overall curvature
(hot – cold)
With the rule astigmatism
(figure 8 pattern)
Correlation

94. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
(cured by pits)

95. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
(cured by pits)
(recall above saw 11 / 25 not conclusive,
so now suppose stronger evidence)

96. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So

97. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So

98. (more conclusive than before)
Hypothesis Testing
H0: p < vs H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So
(more conclusive than before)

99. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So
(more conclusive than before)
(how much stronger is the evidence?)

100. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So
p-value = P[ X ≥ 13 | p = 0.6]

101. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So
p-value = P[ X ≥ 13 | p = 0.6] = 0.027

102. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15 So
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
Calculated similar to above:

103. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
p-value = P[ X ≥ 13 | p = 0.6] = 0.027

104. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
Conclusions:
Yes-No: < 0.05, so can reject H0 and make firm conclusion pits are better

105. Hypothesis Testing H0: p < 0.6 vs. H1: p ≥ 0.6
Now suppose X had been 13 out of 15
p-value = P[ X ≥ 13 | p = 0.6] = 0.027
Conclusions:
Yes-No: < 0.05, so can reject H0 and make firm conclusion pits are better
Gray Level: Strong case, nearly very strong that pits are better

106. Hypothesis Testing In General: p-value = P[what was seen,
or more conclusive | at boundary between H0 & H1]

107. Hypothesis Testing In General: p-value = P[what was seen,
or more conclusive | at boundary between H0 & H1]
(will use this throughout the course,
well beyond Binomial distributions)

108. Hypothesis Testing
HW C14: Answer from both gray-level and yes-no viewpoints:
(a) A TV ad claims that less than 40% of people prefer Brand X. Suppose 7 out of 10 randomly selected people prefer Brand X. Should we dispute the claim? (p-value = 0.055)

109. Hypothesis Testing
HW C14: Answer from both gray-level and yes-no viewpoints:
(b) 80% of the sheet metal we buy from supplier A meets our specs. Supplier B sends us 12 shipments, and 11 meet our specs. Is it safe to say the quality of B is higher? (p-value = 0.275)

110. Warning
Avoid the “Excel Twiddle Trap”

111. Warning
Avoid the “Excel Twiddle Trap”, E.g. C14(a)

112. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Find what Excel needs:

113. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a)
Find what Excel needs:
Number_s:
Trials:
Probability_s:
Cumulative: true
(plug in)

114. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a) Check given answer
(0.055)

115. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a) Check given answer
(0.055)
Way off!

116. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a) Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)

117. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a) Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
Still off, how about
the “> vs. ≥” issue?

118. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a) Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
Still off, how about
the “> vs. ≥” issue?
try replacing 7 by 6?

119. Warning Avoid the “Excel Twiddle Trap”, E.g. C14(a) Check given answer
(0.055)
Way off! Try “1 -”
i.e. target (0.945)
Still off, how about
the “> vs. ≥” issue?
try replacing 7 by 6? Yes!

120. Warning
Avoid the “Excel Twiddle Trap”:
Can solve HW OK

121. Warning Avoid the “Excel Twiddle Trap”: Can solve HW OK
But not on exam
No numerical answer given
No interaction with Excel

122. Warning Avoid the “Excel Twiddle Trap”: Can solve HW OK
But not on exam
No numerical answer given
No interaction with Excel
Real Goal: Understanding Principles

123. And now for something completely different
Lateral Thinking: What is the phrase?

124. And now for something completely different
Lateral Thinking: What is the phrase?
Card
Shark

125. And now for something completely different
Lateral Thinking:
What is the
phrase?

126. And now for something completely different
Lateral Thinking:
What is the
phrase?
Knight Mare

127. And now for something completely different
Lateral Thinking:
What is the
phrase?

128. And now for something completely different
Lateral Thinking:
What is the
phrase?
Gator Aide

129. Hypothesis Testing In General: p-value = P[what was seen,
or more conclusive | at boundary between H0 & H1]

130. Hypothesis Testing In General: p-value = P[what was seen,
or more conclusive | at boundary between H0 & H1]
Caution: more conclusive requires careful interpretation

131. Hypothesis Testing
Caution: more conclusive requires careful interpretation

132. Hypothesis Testing
Caution: more conclusive requires careful interpretation
Reason: Need to decide between
1 - sided Hypotheses

133. Hypothesis Testing
Caution: more conclusive requires careful interpretation
Reason: Need to decide between
1 - sided Hypotheses, like
H0 : p < vs. H1: p ≥
some given numerical value

134. Hypothesis Testing
Caution: more conclusive requires careful interpretation
Reason: Need to decide between
1 - sided Hypotheses, like
H0 : p < vs. H1: p ≥
And 2 - sided Hypotheses

135. Hypothesis Testing
Caution: more conclusive requires careful interpretation
Reason: Need to decide between
1 - sided Hypotheses, like
H0 : p < vs. H1: p ≥
And 2 - sided Hypotheses, like
H0 : p = vs. H1: p ≠

136. Hypothesis Testing 2 - sided Hypotheses, like H0 : p = vs. H1: p ≠
Note: Can never have H1: p =

137. Hypothesis Testing 2 - sided Hypotheses, like H0 : p = vs. H1: p ≠
Note: Can never have H1: p = ,
since can’t tell for sure between
and

138. (Recall: H1 has burden of proof)
Hypothesis Testing
2 - sided Hypotheses, like
H0 : p = vs. H1: p ≠
Note: Can never have H1: p = ,
since can’t tell for sure between
and
(Recall: H1 has burden of proof)

139. Hypothesis Testing
Caution: more conclusive requires careful interpretation
1 - sided Hypotheses & sided Hypotheses

140. (important choice will need to make a lot)
Hypothesis Testing
Caution: more conclusive requires careful interpretation
1 - sided Hypotheses & sided Hypotheses
(important choice will need to make a lot)

141. Hypothesis Testing
Caution: more conclusive requires careful interpretation
1 - sided Hypotheses & sided Hypotheses
Useful Rule: set up 2-sided when problem uses words like “equal” or “different”

142. Hypothesis Testing
e.g. a slot machine
Gambling device

143. Hypothesis Testing e.g. a slot machine Gambling device
Players put money in

144. (of quite a lot more money)
Hypothesis Testing
e.g. a slot machine
Gambling device
Players put money in
With (small) probability, win a “jackpot”
(of quite a lot more money)

145. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”

146. (in real life, focus is on “return rate”)
Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
(in real life, focus is on “return rate”)

147. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
(in real life, focus is on “return rate”)
(since people enjoy fewer, but bigger jackpots)

148. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
(in real life, focus is on “return rate”)
(since people enjoy fewer, but bigger jackpots)
(but usually no signs,
since return rate is < 0)

149. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any.

150. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any.
Can I conclude sign is false?

151. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any.
Can I conclude sign is false?
(& thus have grounds for complaint,
or is this a reasonable occurrence?)

152. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win]

153. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win]
(usual approach: give unknowns a
name, so can work with)

154. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays

155. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)

156. (set up as H0, the point want to disprove)
Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
(set up as H0, the point want to disprove)

157. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = vs. H1: p ≠ 0.3

158. (“false” means don’t win 30% of time,
Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = vs. H1: p ≠ 0.3
(“false” means don’t win 30% of time,
so go 2-sided)

159. Hypothesis Testing Aside (similar to above):
Can never set up H0: p ≠ 0.3

160. Hypothesis Testing Aside (similar to above):
Can never set up H0: p ≠ 0.3
And then prove that p = 0.3

161. Hypothesis Testing Aside (similar to above):
Can never set up H0: p ≠ 0.3
And then prove that p = 0.3
Since can’t handle gray area of hypo test

162. Hypothesis Testing Aside (similar to above):
Can never set up H0: p ≠ 0.3
And then prove that p = 0.3
Since can’t handle gray area of hypo test
E.g. can’t distinguish from p =

163. Hypothesis Testing Aside (similar to above):
Can never set up H0: p ≠ 0.3
And then prove that p = 0.3
Since can’t handle gray area of hypo test
E.g. can’t distinguish from p =
Could always be “off a little bit”

164. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = vs. H1: p ≠ 0.3

165. (now test & see how weird X = 0 is, for p = 0.3)
Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = vs. H1: p ≠ 0.3
(now test & see how weird X = 0 is, for p = 0.3)

166. Hypothesis Testing
e.g. a slot machine bears a sign which says “Win 30% of the time”
In 10 plays, I don’t win any. Conclude false?
Let p = P[win], let X = # wins in 10 plays
Model: X ~ Bi(10, p)
Test: H0: p = vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]

167. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]

168. (understand this by visualizing # line)
Hypothesis Testing
Test: H0: p = vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
(understand this by visualizing # line)

169. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]

170. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
30% of 10, most likely when p = 0.3
i.e. least conclusive

171. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
so more conclusive includes

172. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3
p-value = P[X = 0 or more conclusive | p = 0.3]
so more conclusive includes
but since 2-sided, also include

173. Hypothesis Testing
Generally how to calculate?

174. Hypothesis Testing Generally how to calculate? Observed Value

175. Hypothesis Testing Generally how to calculate? Observed Value
Most Likely Value

176. Hypothesis Testing Generally how to calculate? Observed Value
Most Likely Value
# spaces = 3

177. Hypothesis Testing Generally how to calculate? Observed Value
Most Likely Value
# spaces = 3
so go 3 spaces in other direct’n

178. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6

179. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]

180. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]

181. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
= P[X ≤ 0] + (1 – P[X ≤ 5])

182. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
= P[X ≤ 0] + (1 – P[X ≤ 5]) =

183. Hypothesis Testing Result: More conclusive means X ≤ 0 or X ≥ 6
p-value = P[X = 0 or more conclusive | p = 0.3]
= P[X ≤ 0 or X ≥ 6 | p = 0.3]
= P[X ≤ 0] + (1 – P[X ≤ 5]) =
Excel result from:

184. Hypothesis Testing
Test: H0: p = vs. H1: p ≠ 0.3
p-value = 0.076

185. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = 0.076
Yes-No Conclusion: > 0.05,
so not safe to conclude “P[win] = 0.3” sign
is wrong, at level 0.05

186. (10 straight losses is reasonably likely)
Hypothesis Testing
Test: H0: p = vs. H1: p ≠ 0.3
p-value = 0.076
Yes-No Conclusion: > 0.05,
so not safe to conclude “P[win] = 0.3” sign
is wrong, at level 0.05
(10 straight losses is reasonably likely)

187. Hypothesis Testing Test: H0: p = 0.3 vs. H1: p ≠ 0.3 p-value = 0.076
Yes-No Conclusion: > 0.05,
so not safe to conclude “P[win] = 0.3” sign
is wrong, at level 0.05
Gray Level Conclusion: in “fuzzy zone”, some evidence, but not too strong