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This guide focuses on analyzing two-way tables using the Chi-Squared test for homogeneity and independence. It covers understanding degrees of freedom (df), interpreting null (H₀) and alternative (Hₐ) hypotheses, and ensuring proper assumptions are met. Through practical examples, we demonstrate how to determine if significant differences exist in various scenarios, such as preferences for TV programs or the relationship between gender and political party affiliation. Gain insights into statistical significance and hypothesis testing in a structured manner.
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AP Statistics 13.2 Inference for Two Way Tables
Learning Objective: • Analyze Two Way Tables Using Chi-Squared Test for Homogeneity and Independence
Expected Counts= • Degrees of freedom (r-1)(c-1) Chi-Squared Test Statistic
Chi-Squared (Homogeneity)- • H₀:the proportion of ________ is the SAME as __________ • Ha: the proportion of ________ is the DIFFERENT than __________
Example 1: Do the boys’ preferences for the following TV programs differ significantly from the girls’ preferences? Use a 5% significance level.
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
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?
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
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.
Chi-Squared (Independence)- • H₀: There is no relationship (association) between ________ and ________. • Ha: There is a relationship (association) between ________ and ________.
Example 3: An SRS of 1000 was taken • Is there a relationship between gender and political parties?
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
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?
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
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
