Final answer:
In applying a chi-square test, the deviation squared divided by expected results measures the discrepancy between observed and expected frequencies, with the test statistic indicating the degree of this discrepancy. The test is right-tailed and requires at least five expected occurrences in each cell. Degrees of freedom vary based on the type of chi-square test being used.
Step-by-step explanation:
In applying a chi-square test, the square of the deviation is divided by the expected results, which is a way to assess how observed data deviates from what was expected under the hypothesis being tested. Specifically, this formulation is part of the calculation for the chi-square test statistic, where each observed count is compared to the corresponding expected count. The purpose is to measure the discrepancy between observed and expected frequencies in categorical data.
Test statistics based on the chi-square distribution are always greater than or equal to zero. Such application tests are typically right-tailed, meaning that if the observed values differ significantly from the expected, the test statistic becomes very large and falls into the right tail of the chi-square distribution curve. This is characteristic of both goodness-of-fit tests and tests of independence.
To calculate the expected frequencies, the formula (row total) × (column total) / total number surveyed is used. The number of degrees of freedom (df) for a chi-square test is dependent on the context: for a test of independence, df is calculated as (number of columns - 1)(number of rows - 1), while for a goodness-of-fit test, df is calculated as (number of categories – 1). An essential condition for the chi-square test is that the expected value for each cell needs to be at least five.