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Ch 15 - Chi-Squared _Categorical Data_

Ournextstepistocomputetheexpectedcountsundertheassumpt

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Unformatted text preview: ears to be the same for the two treatment populations. 215 Test of Independence: Helps us to assess if two discrete (categorical) variables are independent for a population, or if there is an association between the two variables. Test of Independence Scenario: We have one population of interest ‐ say factory workers. Question: Is there a relationship between smoking habits and whether or not a factory worker experiences hypertension? Data: 1 random sample of 180 factory workers, we measure the two variables: X = hypertension status (yes or no) Y = smoking habit (non, moderate, heavy) The table below summarizes the data in terms of the observed counts. Observed Counts: X= Hyper Yes Status No Y= Smoking Mod 36 26 62 Non 21 48 69 Habit Heavy 30 19 49 87 93 180 Get the row and column totals. Note: neither the row nor the column totals were known in advance before measuring hypertension and smoking habit. We only know the overall total of 180. The null hypothesis: H0: There is no association between smoking habit and hypertension status for the population of factory workers. (or The two factors, smoking habit and hypertension status, are independent for the population.) One more mathematical way to write this null hypothesis is: H0: P X i and Y j P( X i) P(Y j ) The null hypothesis looks like: P ( A and B ) P ( A) P ( B ) , which is one definition of independent events, from our previous discussion of independence. 216 Getting back to our FACTORY WORKERS … The two‐way table provides the OBSERVED counts. Our next step is to compute the EXPECTED counts, under the assumption that H 0 is true. The expected counts and the test statistic are found the same way as for the homogeneity test. Cross‐Product Rule: Expected Counts (row total)(column total) Total n Compute and enter these expected counts in the parentheses in the table below. Observed Counts (Expected Counts): X= Hyper Yes Status No Y= Non Smoking Mod Habit Heavy 21 ( 33.35 ) 48 ( 35.65 ) 69 36 ( 29.97 ) 26 ( 32.03 ) 62 30 (23.68 ) 19 (25.32 ) 49 87 93 180 2 The X test sta...
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