Bullied as a child?

Are you tall or short? 6’ 4” 5’ 10” 4’ 2’ 4”

Is a persons’ size related to if they were bullied You gathered data from 209 children at Springfield Elementary School. Assessed: Height (short vs. not short) Bullied (yes vs. no)

Results Ever Bullied

Results Ever Bullied

Results Ever Bullied

Results Ever Bullied

Results Ever Bullied

Results Ever Bullied

Is this difference in proportion due to chance? To test this you use a Chi-Square (  2 ) Notice how this is different than a t-test or an ANOVA –t-tests and ANOVAs use quantitative variables –Chi-squares use frequency counts of categories

Hypothesis H 1: There is a relationship between the two variables –i.e., a persons size is related to if they were bullied H 0 :The two variables are independent of each other –i.e., there is no relationship between a persons size and if they were bullied

Logic Is the same as t-tests and ANOVAs 1) calculate an observed Chi-square 2) Find a critical value 3) See if the the observed Chi-square falls in the critical area

Chi-Square O = observed frequency E = expected frequency

Results Ever Bullied

Observed Frequencies Ever Bullied

Expected frequencies Are how many observations you would expect in each cell if the null hypothesis was true –i.e., there there was no relationship between a persons size and if they were bullied

Expected frequencies To calculate a cells expected frequency: For each cell you do this formula

Expected Frequencies Ever Bullied

Expected Frequencies Ever Bullied

Expected Frequencies Ever Bullied Row total = 92

Expected Frequencies Ever Bullied Row total = 92 Column total = 72

Expected Frequencies Ever Bullied Row total = 92 N = 209 Column total = 72

Expected Frequencies Ever Bullied E = (92 * 72) /209 = 31.69

Expected Frequencies Ever Bullied

Expected Frequencies Ever Bullied

Expected Frequencies Ever Bullied E = (92 * 137) /209 = 60.30

Expected Frequencies Ever Bullied E = (117 * 72) / 209 = E = (117 * 137) / 209 = 76.69

Expected Frequencies Ever Bullied The expected frequencies are what you would expect if there was no relationship between the two variables!

How do the expected frequencies work? Ever Bullied Looking only at:

How do the expected frequencies work? Ever Bullied If you randomly selected a person from these 209 people what is the probability you would select a person who is short?

How do the expected frequencies work? Ever Bullied If you randomly selected a person from these 209 people what is the probability you would select a person who is short? 92 / 209 =.44

How do the expected frequencies work? Ever Bullied If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied?

How do the expected frequencies work? Ever Bullied If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied? 72 / 209 =.34

How do the expected frequencies work? Ever Bullied If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied and is short?

How do the expected frequencies work? Ever Bullied If you randomly selected a person from these 209 people what is the probability you would select a person who was bullied and is short? (.44) (.34) =.15

How do the expected frequencies work? Ever Bullied How many people do you expect to have been bullied and short?

How do the expected frequencies work? Ever Bullied How many people would you expect to have been bullied and short? (.15 * 209) = (difference due to rounding)

Back to Chi-Square O = observed frequency E = expected frequency

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22

22

22

22

22

22

Significance Is a  2 of 9.13 significant at the.05 level? To find out you need to know df

Degrees of Freedom To determine the degrees of freedom you use the number of rows (R) and the number of columns (C) DF = (R - 1)(C - 1)

Degrees of Freedom Ever Bullied Rows = 2

Degrees of Freedom Ever Bullied Rows = 2 Columns = 2

Degrees of Freedom To determine the degrees of freedom you use the number of rows (R) and the number of columns (C) df = (R - 1)(C - 1) df = (2 - 1)(2 - 1) = 1

Significance Look on Table E -- page 389 df = 1  =.05  2 critical = 3.84

Decision Thus, if  2 > than  2 critical –Reject H 0, and accept H 1 If  2 < or = to  2 critical –Fail to reject H 0

Current Example  2 = 9.13  2 critical = 3.84 Thus, reject H 0, and accept H 1

Current Example H 1: There is a relationship between the the two variables –A persons size is significantly (alpha =.05) related to if they were bullied

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 H 1: There is a relationship between gender and with whom a person finds it easiest to make friends H 0 :Gender and with whom a person finds it easiest to make friends are independent of each other

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 in your data table

Step 4: Calculate the Expected Frequencies E = (73 * 137) /205 = 48.79

Step 4: Calculate the Expected Frequencies E = (73 * 68) /205 = 24.21

Step 4: Calculate the Expected Frequencies E = (29 * 137) /205 = 19.38

Step 4: Calculate the Expected Frequencies