We can also compute each count as a percent of the column total These percents

# We can also compute each count as a percent of the

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We can also compute each count as a percent of the column total. These percents should add up to 100% and together are the conditional distribution.

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Formulating your hypothesis Like in previous chapters, we want to know if the differences in sample proportions are likely to have occurred just by chance, because of the random sampling. Like in previous chapters, we formulate our null- hypothesis in terms of no association (relation) between the categorical variables. We use the chi-square ( c 2 ) test to assess the null hypothesis of no relationship between the two categorical variables of a two-way table.
The chi-square statistic ( c 2 ) is a measure of how much the observed cell counts in a two-way table diverge from the expected cell counts. The formula for the c 2 statistic is: Large values for c 2 represent strong deviations from the expected distribution under the H 0 , providing evidence against H 0 . Note that, since c 2 is a sum, how large a c 2 is required for The chi-square test c 2 observed count - expected count 2 expected count

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Expected counts in two-way tables We want to test the hypothesis that there is no relationship between these two categorical variables ( H 0 ). To test this hypothesis, we compare actual counts from the sample data with expected counts , given the null hypothesis of no relationship. The expected count in any cell of a two-way table when H 0 is true is:
The Chi-square test For the chi-square test, H 0 states that there is no association between the row and column variables in a two-way table. The alternative is that these variables are related. If H 0 is true, the chi-square test has approximately a χ 2 distribution with ( r − 1)( c − 1) degrees of freedom . The P -value for the chi- square test is the area to the right of c 2 under the c 2 distribution with df (r−1) (c−1): P( c 2 X 2 ).

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Chi-square test vs. z test for two proportions When comparing only two proportions, such as in a 2x2 table where the columns represent counts of “success” and “failure,” we can test H 0 : p 1 = p 2 vs. H a p 1 p 2

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