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

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?

Results

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

Step 2: Create the Data Table

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

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

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

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

Step 4: Calculate the Expected Frequencies

Step 4: Calculate the Expected Frequencies

Step 4: Calculate the Expected Frequencies

Step 4: Calculate the Expected Frequencies

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

2

2

2

2

2 8.5

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

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

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.

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?”

Results Marijuana ? Death Penalty ?

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

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

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

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

Step 5: Calculate 2

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

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

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

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.

Phi Use with 2x2 tables N = sample size

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?”

Results Marijuana ? Death Penalty ?

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

Phi Use with 2x2 tables

Bullied Example Ever Bullied

2

Phi Use with 2x2 tables

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?

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

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

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

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

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?

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

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

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

Step 3: Create the data table

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

Step 4: Calculate the Expected Frequencies

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

Step 4: Calculate the Expected Frequencies Expected Frequency = (.30)(2081) = 624.30

Step 4: Calculate the Expected Frequencies Expected Frequency = (.20)(2081) = 416.20

Step 4: Calculate the Expected Frequencies Expected Frequency = (.20)(2081) = 416.20

Step 4: Calculate the Expected Frequencies Expected Frequency = (.10)(2081) = 208.10

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

2

2

2

2

2

2 15.52

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

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

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

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

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)?

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

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

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

Step 3: Create the data table

Step 4: Calculate the Expected Frequencies

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

2 19.42

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

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

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)

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

Results X2 = 5.38 X2 crit = 3.83 English ADD

Phi Use with 2x2 tables

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?

Practice p zero will accept = .11 p one will accept = .27 p zero OR one will accept = .38 p two or more will accept = 1 - .38 = .62

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?

Step 1: Sketch out question 95 110 ? -3 -2 -1  1 2  3 

Step 2: Calculate Z scores for both values Z = (X -  ) /  Z = (95 - 100) / 15 = -.33 Z = (110 - 100) / 15 = .67

Step 3: Look up Z scores -.33 .67 .1293 .2486 -3 -2 -1  1 2  3 

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

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

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.

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

Practice A = 32 28 B = 61 52 C = 64 62 D = 31 38 F = 12 20 Chi square = 6.68 Critical Chi square (4) = 9.49

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

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

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.

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

http://www.ds.unifi.it/VL/VL_EN/bernoulli/bernoulli2.html

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.

Binomial Distribution p = .67 p Aces

Binomial Distribution p = .62 p Aces

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!

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.