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Assuming independence allows researchers to use the multiplication rule for independent events to calculate probabilities. When two variables are independent, they do not influence each other, so the probability of any event involving both variables equals the product of their individual probabilities. This assumption is fundamental to determining expected frequencies in a contingency table for conducting a test of independence.

The expected frequency for each cell is calculated by multiplying the probability of the event by the grand total of the contingency table. This can be simplified to: expected frequency equals the row total multiplied by the column total, divided by the grand total. This formula is applied consistently across all cells to obtain complete expected frequencies for the dataset.

Each expected frequency in a contingency table must be at least 5. This requirement ensures the validity and reliability of statistical tests performed on the data. If expected frequencies fall below this threshold, the results of hypothesis tests may be unreliable or invalid.

Observed frequencies are the actual values recorded in the contingency table from collected data. Expected frequencies are theoretical values calculated assuming the two variables are independent. Comparing these two sets of frequencies allows researchers to determine whether the variables are truly independent or if a significant relationship exists between them.

After calculating expected frequencies, researchers determine the chi-square test statistic by comparing observed and expected frequencies. This statistic is then used to conduct a hypothesis test and calculate the P-value, which determines whether there is sufficient evidence to reject the independence assumption and conclude that the variables are related.

The multiplication rule for independent events states that the probability of both events occurring equals the product of their individual probabilities. For expected frequencies, this rule means multiplying the individual probability values from the contingency table, then multiplying by the grand total to obtain the expected frequency for each cell.

Expected frequencies provide the theoretical baseline needed to calculate the chi-square test statistic and assess whether observed data significantly deviates from independence. Without expected frequencies, researchers cannot quantify the difference between actual and theoretical distributions or determine the P-value required for hypothesis testing.