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AP Statistics13.2 Inference for Two Way Tables
Analyze Two Way Tables Using Chi-Squared Test for Homogeneity and Independence
Learning Objective:
Goodness of Fit
Homogeneity
Independence
1 variable
-distribution
2 variables (2 way table) -distribution-proportions
2 variables (2 way table)
-association-dependent upon-relationship-influences
Three Types of Chi-Squared Distributions
Is there evidence the wheel is unbalanced?
(one variable- prize you get)
Example of Chi-Squared(Goodness of Fit) Test
Prize Teddy Bear
Goldfish T-shirt Poster
# of winners
34 41 39 27
Is there evidence that proportion of kids who attend regular MGS is different for each grade level.
Notice it is a 2-way table with two categorical variables:
Grade level and whether you attend MGS
Example of Chi-Squared(Homogeneity) Test
Freshmen
Sophomore
Junior Senior
Attend MGS
45 61 99 81
Does Not
Attend MGS
97 86 64 72
This is the same thing as a test of homogeneity except the wording of the question will use key words such as association or relationship. We complete the same steps in our calculator, we just use a different name for our test and word our Ho and Ha different.
Is there evidence of a relationship between grade level and kids who attend MGS regularly?
Example of Chi-Squared(Independence) Test
Expected Counts=
Degrees of freedom
(r-1)(c-1)
Chi-Squared Test Statistic
totaltable
totalcolumntotalrow
Find the expected counts: First you need to find the row and column totals above. I did the first example-you fill in the rest!
Expected Freshmen Sophomore Junior Senior
Attend MGS (286*142)/605=67.13
Does Not Attend MGS
Observed
Freshmen
Sophomore
Junior
Senior
Total
Attend MGS
45 61 99 81 286
Does Not Attend MGS
97 86 64 72 319
Total 142 147 163 153 605
Expected counts complete! Now let’s try the quick way. Follow along!
Go to your calculator. Matrix-Edit-enter [A] Matrix[A] r x c (so change it to 2 x 4). Then input your observed
counts. Then hit: stat-tests-x² test. Just hit calculate. It gives you your calculations but we can worry about those later!!
Go back to matrix and hit enter on matrix [B]. When you hit enter again scroll through the matrix and notice your
calculator did all the expected counts and they should match what you just did by hand!
Expected Freshmen Sophomore Junior Senior
Attend MGS (286*142)/605=67.13
69.49 77.05 72.32
Does Not Attend MGS
74.87 77.51 85.95 80.67
X²-testObserved:[A]Expected: [B]Calculate
df =(r-1)(c-1) =(2-1)(4-1)
=1*3=3
Degrees of freedom
Observed
Freshmen
Sophomore
Junior
Senior
Total
Attend MGS
45 61 99 81 286
Does Not Attend MGS
97 86 64 72 319
Total 142 147 163 153 605
H₀:the proportion of ________ is the SAME as __________
Ha: the proportion of ________ is DIFFERENT than __________
(these are just template sentences, remember whatever the question is asking is your Ha)
Chi-Squared (Homogeneity)-
Example 1: Do the boys’ preferences for the following TV programs differ significantly from the girls’ preferences? Use a 5% significance level.
House Grey’sAnatomy
AmericanIdol
CSI
Boys 66 78 67 105
Girls 48 130 123 61
H₀:the boys preference for TV programs is the SAME as the girls
Ha: the boys preference for TV programs is DIFFERENT than the girls
Assumptions:-random sample-all expected counts are ≥ 1-no more than 20% of the expected counts
<5 House Grey’sAnatomy
AmericanIdol
CSI
Boys 53.1 96.9 88.6 77.4
Girls 60.9 111.1 101.4 88.6
Chi-Squared Test (Homogeneity) w/ α=0.05
P(x²>41.08)=0.000000006 df=3
Since p< α, it is statistically significant. Therefore we reject H₀. There is enough evidence to say the preference of TV programs for boys is different than girls.
Example 2: The following data is an SRS of 650 patients at a local hospital. Does the effect of aspirin significantly differ from a placebo for these medical conditions?
Aspirin Placebo
Fatal Heart Attacks
20 60
Non-Fatal Heart Attacks
125 220
Strokes 75 150
H₀:the effects of aspirin is the same as the placebo
Ha: the effects of aspirin is different than the placebo
Assumptions:-random sample-all expected counts are ≥ 1-no more than 20% of the expected counts
<5 Aspirin Placebo
Fatal Heart Attacks
27.1 52.9
Non-Fatal Heart Attacks
116.8 228.2
Strokes 76.2 148.8
Chi-Squared Test (Homogeneity) w/ α=0.05
P(x²>3.70)=0.1573 df=2
Since p∡ α, it is not statistically significant. Therefore we do not reject H₀. There is not enough evidence to say the effect of aspirin differs from the placebo.
H₀: There is no relationship (association) between ________ and ________.
Ha: There is a relationship (association) between ________ and ________.
Chi-Squared (Independence)-
Example 3: An SRS of 1000 was taken
Is there a relationship between gender and political parties?
Republican Democrat Independent
Male 200 150 50
Female 250 300 50
H₀: There is no relationship between gender and political party
Ha: There is a relationship between gender and political party
Assumptions:-random sample-all expected counts are ≥ 1-no more than 20% of the expected counts
<5 Republican Democrat Independent
Male 180 180 40
Female 270 270 60
Chi-Squared Test (Independence) w/ α=0.05
P(x²>16.2)=0.0003 df=2
Since p< α, it is statistically significant. Therefore we reject H₀. There is enough evidence to say there is a relationship between gender and political party
Example 4: An SRS of 592 people were taken comparing their hair and eye color.
Is there an association between hair color and eye color?
Black Brown Red Blonde
Brown 68 119 26 7
Green 20 84 17 94
Blue 15 54 14 10
Hazel 8 29 14 16
H₀: There is no association between hair color and eye color
Ha: There is an association between hair color and eye color
Assumptions:-random sample-all expected counts are ≥ 1-no more than 20% of the expected counts
<5 Black Brown Red Blonde
Brown 41.0 105.7 26.3 47.0
Green 40.1 103.3 25.7 45.9
Blue 17.3 44.7 11.1 19.9
Hazel 12.5 32.2 8.0 14.3
Chi-Squared Test (Independence) w/ α=0.05
P(x²>134.98)≈0 df=9
Since p< α, it is statistically significant. Therefore we reject H₀. There is enough evidence to say there is an association between hair color and eye color
