卡方檢定：練習題

A contingency table is used to test the association between two factors. Factor 1 has 5 levels and Factor 2 has 4 levels. In the calculation of the \( \chi^2 \) test statistic, it is found that one expected frequency is less than 5, so two adjacent rows are pooled together. What are the degrees of freedom for the \( \chi^2 \) test after this pooling?

A contingency table with 3 rows and 3 columns is used to test independence. If the expected frequency in one cell is 3.5, and this cell is merged with an adjacent cell in the same row, what is the number of degrees of freedom for the resulting \( \chi^2 \) test?

A researcher uses a \( \chi^2 \) goodness-of-fit test to check if a die is fair. The die is rolled 120 times. The calculated value of the test statistic is \( \chi^2 = 12.5 \). At a significance level of \( 5\% \), the critical value is \( \chi^2_{5, 0.05} = 11.07 \). At a significance level of \( 1\% \), the critical value is \( \chi^2_{5, 0.01} = 15.09 \). Which of the following is the correct conclusion?

A genetics experiment classifies 200 offspring into four categories: A, B, C, and D. According to a model, the probabilities are \( 0.4, 0.3, 0.2, \) and \( 0.1 \) respectively. The observed frequencies are 70, 65, 45, and 20. Calculate the \( \chi^2 \) goodness-of-fit test statistic.

A researcher is testing the independence of two categorical variables using a contingency table with 5 rows and 4 columns. Calculate the number of degrees of freedom for the \(\chi^2\)-test.

A \(\chi^2\) test for independence is carried out on a \(4 \times 5\) contingency table. In the process, two rows are pooled due to low expected frequencies, and subsequently, three columns are pooled into one. Calculate the final degrees of freedom for the test statistic.