If you are familiar with Excel, you can create a table of the actual counts and a table of the expected counts, and then use the command CHISQ.TEST(actual_range,expected_range), to calculate x 2-test statistic. Next we calculate the x 2-test statistic. Can you conclude that the proportions of workers with the various symptoms differ among shifts? Following are the numbers of workers reporting various symptoms, categorized by work shift. Yalin, Journal of Environmental Science and Health, 2004: 1129-1139) presents assessments of health outcomes of people working in an environment with high levels of carbon monoxide (CO). The article “Determination of Carboxyhemoglobin Levels and Health Effect on Officers Working at the Istanbul Bosphorus Bridge”(G. The graph below is how the distribution of chi-squared looks for three different values of k. Since there are two categorical variables to consider our degrees of freedom is k = ( r – 1)( c – 1), where r and c are the number of rows and columns, respectively. For this reason calculating p-values on this particular distribution is best done using technology or tables. The chi-squared distribution is different depending on the degrees of freedom present in our data, just like was true for the Student’s t-distribution. However, small values of x 2 are not evidence against the null hypothesis. As such, any violation of the null hypothesis will produce a large test statistic. Be careful, as we will see later the x 2distribution is not symmetric, and the alternative hypothesis has many options for sides and directions but chi is one-sided. This will give us evidence to suggest that our null hypothesis is not true. This tells us that large values of the test statistic will indicate that the values are far apart, or rather that the distributions are different. Returning to our example, our test statistic is: The test statistic for the chi-squared test is a measure of how far apart the observed values are from the expected in all cells of the two-way tables.
The use of the chi-squared test statistic is only appropriate when all of the expected counts are greater than or equal to 5! The x 2 compares the observed outcomes with the outcomes expected by assuming a null hypothesis H 0 is true.
The x 2-test is an extension of this concept to a multinomial trials where there are k outcomes, each with probability of success p 1, p 2, …, p k. If one thinks back to the problems that considered the difference of two proportions, the method considered a binomial variables, with probability of success p and compared this with another proportion in order to find a statistically significant difference.
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