2. Seven Steps for Doing 2 1) State the hypothesis 2) Create data table
3) Find 2 critical
4) Calculate the expected frequencies
5) Calculate 2
6) Decision
7) Put answer into words

3. Example With whom do you find it easiest to make friends?
Subjects were either male and female.
Possible responses were: “opposite sex”, “same sex”, or “no difference”
Is there a significant (.05) relationship between the gender of the subject and their response?

4. Results

5. Step 1: State the Hypothesis
H1: There is a relationship between gender and with whom a person finds it easiest to make friends
H0:Gender and with whom a person finds it easiest to make friends are independent of each other

6. Step 2: Create the Data Table

7. Step 2: Create the Data Table
Add “total” columns and rows

8. Step 3: Find 2 critical
df = (R - 1)(C - 1)

9. Step 3: Find 2 critical df = (R - 1)(C - 1) df = (2 - 1)(3 - 1) = 2
 = .05
2 critical = 5.99

10. Step 4: Calculate the Expected Frequencies
Two steps:
4.1) Calculate values
4.2) Put values on your data table

11. Step 4: Calculate the Expected Frequencies

12. Step 4: Calculate the Expected Frequencies

13. Step 4: Calculate the Expected Frequencies

14. Step 4: Calculate the Expected Frequencies

15. Step 5: Calculate 2
O = observed frequency
E = expected frequency

16. 2

17. 2

18. 2

19. 2

20. 2
8.5

21. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

22. Step 6: Decision Thus, if 2 > than 2critical
2 = 8.5
2 crit = 5.99
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

23. Step 7: Put it answer into words
H1: There is a relationship between gender and with whom a person finds it easiest to make friends
A persons gender is significantly (.05) related with whom it is easiest to make friends.

25. Practice
Is there a significant ( = .01) relationship between opinions about the death penalty and opinions about the legalization of marijuana?
933 Subjects responded yes or no to:
“Do you favor the death penalty for persons convicted of murder?”
“Do you think the use of marijuana should be made legal?”

26. Results
Marijuana ?
Death Penalty ?

27. Step 1: State the Hypothesis
H1: There is a relationship between opinions about the death penalty and the legalization of marijuana
H0:Opinions about the death penalty and the legalization of marijuana are independent of each other

28. Step 2: Create the Data Table
Marijuana ?
Death Penalty ?

29. Step 3: Find 2 critical df = (R - 1)(C - 1) df = (2 - 1)(2 - 1) = 1
 = .01
2 critical = 6.64

30. Step 4: Calculate the Expected Frequencies
Marijuana ?
Death Penalty ?

31. Step 5: Calculate 2

32. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

33. Step 6: Decision Thus, if 2 > than 2critical
2 = 3.91
2 crit = 6.64
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

34. Step 7: Put it answer into words
H0:Opinions about the death penalty and the legalization of marijuana are independent of each other
A persons opinion about the death penalty is not significantly (p > .01) related with their opinion about the legalization of marijuana

35. Effect Size Chi-Square tests are null hypothesis tests
Tells you nothing about the “size” of the effect
Phi (Ø)
Can be interpreted as a correlation coefficient.

36. Phi
Use with 2x2 tables
N = sample size

37. Practice
Is there a significant ( = .01) relationship between opinions about the death penalty and opinions about the legalization of marijuana?
933 Subjects responded yes or no to:
“Do you favor the death penalty for persons convicted of murder?”
“Do you think the use of marijuana should be made legal?”

38. Results
Marijuana ?
Death Penalty ?

39. Step 6: Decision Thus, if 2 > than 2critical
2 = 3.91
2 crit = 6.64
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

40. Phi
Use with 2x2 tables

41. Bullied Example
Ever Bullied

42. 2

43. Phi
Use with 2x2 tables

45. 2 as a test for goodness of fit
But what if:
You have a theory or hypothesis that the frequencies should occur in a particular manner?

46. Example M&Ms claim that of their candies: 30% are brown 20% are red
20% are yellow
10% are blue
10% are orange
10% are green

47. Example
Based on genetic theory you hypothesize that in the population:
45% have brown eyes
35% have blue eyes
20% have another eye color

48. To solve you use the same basic steps as before (slightly different order)
1) State the hypothesis
2) Find 2 critical
3) Create data table
4) Calculate the expected frequencies
5) Calculate 2
6) Decision
7) Put answer into words

49. Example M&Ms claim that of their candies: 30% are brown 20% are red
20% are yellow
10% are blue
10% are orange
10% are green

50. Example Four 1-pound bags of plain M&Ms are purchased
Each M&Ms is counted and categorized according to its color
Question: Is M&Ms “theory” about the colors of M&Ms correct?

52. Step 1: State the Hypothesis
H0: The data do fit the model
i.e., the observed data does agree with M&M’s theory
H1: The data do not fit the model
i.e., the observed data does not agree with M&M’s theory
NOTE: These are backwards from what you have done before

53. Step 2: Find 2 critical
df = number of categories - 1

54. Step 2: Find 2 critical df = number of categories - 1 df = 6 - 1 = 5
 = .05
2 critical = 11.07

55. Step 3: Create the data table

56. Step 3: Create the data table
Add the expected proportion of each category

57. Step 4: Calculate the Expected Frequencies

58. Step 4: Calculate the Expected Frequencies
Expected Frequency = (proportion)(N)

59. Step 4: Calculate the Expected Frequencies
Expected Frequency = (.30)(2081) =

60. Step 4: Calculate the Expected Frequencies
Expected Frequency = (.20)(2081) =

61. Step 4: Calculate the Expected Frequencies
Expected Frequency = (.20)(2081) =

62. Step 4: Calculate the Expected Frequencies
Expected Frequency = (.10)(2081) =

63. Step 5: Calculate 2
O = observed frequency
E = expected frequency

64. 2

65. 2

66. 2

67. 2

68. 2

69. 2
15.52

70. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

71. Step 6: Decision Thus, if 2 > than 2critical
2 = 15.52
2 crit = 11.07
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

72. Step 7: Put it answer into words
H1: The data do not fit the model
M&M’s color “theory” did not significantly (.05) fit the data

73. Practice Among women in the general population under the age of 40:
60% are married
23% are single
4% are separated
12% are divorced
1% are widowed

74. Practice You sample 200 female executives under the age of 40
Question: Is marital status distributed the same way in the population of female executives as in the general population ( = .05)?

76. Step 1: State the Hypothesis
H0: The data do fit the model
i.e., marital status is distributed the same way in the population of female executives as in the general population
H1: The data do not fit the model
i.e., marital status is not distributed the same way in the population of female executives as in the general population

77. Step 2: Find 2 critical
df = number of categories - 1

78. Step 2: Find 2 critical df = number of categories - 1 df = 5 - 1 = 4
 = .05
2 critical = 9.49

79. Step 3: Create the data table

80. Step 4: Calculate the Expected Frequencies

81. Step 5: Calculate 2
O = observed frequency
E = expected frequency

82. 2
19.42

83. Step 6: Decision Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

84. Step 6: Decision Thus, if 2 > than 2critical
2 = 19.42
2 crit = 9.49
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

85. Step 7: Put it answer into words
H1: The data do not fit the model
Marital status is not distributed the same way in the population of female executives as in the general population ( = .05)

86. Practice
Practice
Page 169 #6.8
How strong is the relationship?

87. Results
X2 = 5.38
X2 crit = 3.83
English
ADD

88. Phi
Use with 2x2 tables

90. Practice
In the past you have had a 20% success rate at getting someone to accept a date from you.
What is the probability that at least 2 of the next 10 people you ask out will accept?

91. Practice p zero will accept = .11 p one will accept = .27
p zero OR one will accept = .38
p two or more will accept = = .62

93. Practice IQ
Mean = 100
SD = 15
What is the probability that the stranger you just bumped into on the street has an IQ between 95 and 110?

94. Step 1: Sketch out question95
110 ?
-3 -2 -1  1 2  3 

95. Step 2: Calculate Z scores for both values
Z = (X -  ) / 
Z = ( ) / 15 = -.33
Z = ( ) / 15 = .67

96. Step 3: Look up Z scores
-.33
.67
-3 -2 -1  1 2  3 

97. Step 4: Add together the two values
-.33
.67
.3779
-3 -2 -1  1 2  3 

99. Practice
A professor would like to determine if there has been a change in grading practices over the years. In the past, the overall grade distribution was 14% As, 26% Bs, 31% Cs, 19% Ds, and 10% Fs.
A sample of 200 students this years had the following grades

100. Practice
A = 32
B = 61
C = 64
D = 31
F = 12
Do the data indicate a significant change in the grade distribution? Test at the .05 level.

101. Step 1: State the Hypothesis
H0: The data do fit the model
i.e., the grades are distributed the same
H1: The data do not fit the model
i.e., the grades are not distributed the same

102. Practice
A =
B =
C =
D =
F =
Chi square = 6.68
Critical Chi square (4) = 9.49

103. Step 6: Decision Thus, if 2 > than 2critical
2 = 6.68
2 crit = 9.49
Thus, if 2 > than 2critical
Reject H0, and accept H1
If 2 < or = to 2critical
Fail to reject H0

104. Step 7 H0: The data do fit the model
i.e., the grades are distributed the same
There is no evidence that the grades have changed

105. Practice
In the 1930’s 650 boys participated in the Cambridge-Somerville Youth Study. Half of the participants were randomly assigned to a delinquency-prevention pogrom and the other half to a control group. At the end of the study, police records were examined for evidence of delinquency. In the prevention program 114 boys had a police record and in the control group 101 boys had a police record. Analyze the data and write a conclusion.

106. Chi Square observed = 1.17 Chi Square critical = 3.84 Phi = .04
Note the results go in the opposite direction that was expected!

110. Practice
In 1693, Samuel Pepys asked Isaac Newton whether it is more likely to get at least one ace in 6 rolls of a die or at least two aces in 12 rolls of a die. This problems is known a Pepys' problem.

111. Binomial Distribution
p = .67 p
Aces

112. Binomial Distribution
p = .62 p
Aces

113. Practice
In 1693, Samuel Pepys asked Isaac Newton whether it is more likely to get at least one ace in 6 rolls of a die or at least two aces in 12 rolls of a die. This problems is known a Pepys' problem.
It is more likely to get at least one ace in 6 rolls of a die!

115. Practice
Which is more likely: at least one ace with 4 throws of a fair die or at least one double ace in 24 throws of two fair dice? This is known as DeMere's problem, named after Chevalier De Mere.
Blaise Pascal later solved this problem. 

116. Binomial Distribution
p = .482 of zero aces
= .518 at least one ace will occur

117. Binomial Distribution
p = .508 of zero double aces
= .492 at least one double ace will occur

118. Practice
Which is more likely: at least one ace with 4 throws of a fair die or at least one double ace in 24 throws of two fair dice? This is known as DeMere's problem, named after Chevalier De Mere.
More likely at least one ace with 4 throws will occur

119. Extra Brownie Points! Lottery To Win: choose the 5 winnings numbers
from 1 to 49
AND
Choose the "Powerball" number
from 1 to 42
What is the probability you will win?
Use combinations to answer this question