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

Three Types of Chi-Squared Distributions

Expected Counts=

Degrees of freedom

(r-1)(c-1)

Chi-Squared Test Statistic

totaltabletotalcolumntotalrow

H₀:the proportion of ________ is the SAME as __________

Ha: the proportion of ________ is the DIFFERENT than __________

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 105Girls 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 77.4Girls 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 PlaceboFatal 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 7Green 20 84 17 94Blue 15 54 14 10Hazel 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 47.0Green 40.1 103.3 25.7 45.9Blue 17.3 44.7 11.1 19.9Hazel 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