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Ch 15 - Chi-Squared _Categorical Data_

Thenullhypothesis letpiproportionofcarsusingbooth i

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Unformatted text preview: record one variable X, which booth was used (1, 2, 3, 4). The table below summarizes the data in terms of the observed counts. Observed # cars Booth 1 26 Booth 2 20 Booth 3 28 Booth 4 26 Note: This is only a one‐way frequency table, not a two‐way table as will be in the homogeneity and independence tests. We use the notation k = the number of categories or cells, here k 4 . The null hypothesis: Let pi = proportion of cars using booth i H0: p1 = 0.25 , p2 = 0.25 , p3 = 0.25 , p4 = 0.25 . Ha: ___ not all probabilities specified in H0 are correct _______________ The null hypothesis specifies a particular discrete model (mass function) by listing the proportions (or probabilities) for each of the k outcome categories. The one‐way table provides the OBSERVED counts. Our next step is to compute the EXPECTED counts, under the assumption that H0 is true. 207 How to find the expected counts? There were 100 cars in the sample and 4 booths. If the booths are used equally often, H0 is true, then we would expect ... 25 cars to use Booth #1 How did you get the 25? 25% of 100 (np) ... 25 cars to use Booth #2 ... 25 cars to use Booth #3 ... 25 cars to use Booth #4 Expected Counts Ei np i Enter these expected counts in the parentheses in the table below. Observed Counts (Expected Counts)} Booth 1 Booth 2 Booth 3 Number of cars 26 ( 25 ) 20 ( 25 ) 28 ( 25 ) Booth 4 26 ( 25 ) The X 2 test statistic Next we need our test statistic, our measure of how close the observed counts are to what we expect under the null hypothesis. X2 O E 2 26 252 20 252 28 252 26 252 E 25 25 25 25 (1 25 9 1) / 25 36 / 25 1.44 Do you think a value of X 2 1.44 is large enough to reject H0? Let's find the p‐value, the probability of getting an X 2 test statistic value as large or larger than the one we observed, assuming H0 is true. To do this we need to know the distribution of the X 2 test statistic under thenull hypothesis. If H0 is true, then X 2 has the 2 distribution with degrees of freedom = k – 1 . allcells 208 Find the p‐value for our tollbooth example: Obs...
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