Question 2: What conditions do the data have to meet for the test to be valid?
The chi-squared test is a large sample test and the usual rule is that the large sample approximation holds if all expected frequencies are greater than 5 for a 2 by 2 table. Although one observed frequency is 5, no expected values will be as small. This is because if the null hypothesis were true then the overall probability of being positive for P. alcalifaciens would be 28/627 = 0.04 and this proportion would apply to those who have and those who have not travelled abroad. Thus the expected numbers positive for P. alcalifaciens would be 254 × 28/627 = 11.3 for those who have travelled abroad and 373 × 28/627 = 16.7 among those who have not travelled abroad. The other expected values can be calculated in a similar way but will be large because the expected values must add to the marginal totals for each row and column.
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