Final answer:
A chi-square test for independence is used to determine if there is a significant relationship between two categorical variables. Expected frequencies are calculated and used to find the chi-square statistic, which is compared to the critical value at the desired significance level. It's important to ensure all expected frequencies are above 5 to maintain the validity of the test.
Step-by-step explanation:
When conducting a chi-square test for independence, the null hypothesis posits that the two categorical variables are independent of each other. This statistical test is used to determine whether there is a significant association between the two categories represented in a contingency table. If the chi-square statistic is below a critical value from the chi-square distribution table, we fail to reject the null hypothesis, which means there is not enough evidence to say that the variables are not independent. Conversely, if it's above the critical value, we reject the null hypothesis, supporting the claim that there is an association between the variables.
Calculation of expected frequencies helps in determining the chi-square statistic. According to the degrees of freedom, which in a 2x2 matrix is (2-1)(2-1) = 1, we compare the calculated chi-square statistic with the critical value observed from the chi-square distribution table at the desired significance level. The typical level of significance used is 0.05. If a cell's expected frequency is less than 5, the validity of the chi-square test may be compromised; hence it's important to ensure each expected count is above this threshold.
Following the formula for expected frequency (E = (row total x column total) / total surveyed), one can compute the expected counts for each cell of the contingency table, rounding to two decimal places for precision. The chi-square statistic is then calculated by summing the squares of the differences between observed and expected counts, divided by the expected counts for each cell. This process is necessary to assess how well the observed data fits the assumption of independence under the null hypothesis.