Question Type 5: Calculating the chi-squared value and/or p-value to conclude the chi-squared independence test Exercises

The following table shows the observed counts for two age groups across four media usage categories:

A $\chi^2$ test for independence is performed at the $1\%$ significance level.

Calculate the $\chi^2$ test statistic and the $p$-value for this test, and state the conclusion in context, justifying your answer.

A survey records eye colour (Blue, Green, Brown) and blood type (A, B, O). The observed counts are shown in the table below:

Calculate the $\chi^2$ statistic for testing whether eye colour and blood type are independent.

A study on two factors produces the following $2 \times 2$ table of observed counts:

Calculate the $\chi^2$ statistic for testing independence between smoking and disease status.

The following $2 \times 2$ table shows observed frequencies of a treatment vs recovery outcome:

Perform a chi-squared test for independence at the $5\%$ significance level. Calculate the test statistic and state your conclusion.

You conduct a chi-squared test and obtain $\chi^2 = 4.605$ with $1$ degree of freedom. Find the $p$-value and conclude at the $5\%$ significance level.

A $3 \times 3$ contingency table yields $\chi^2 = 12.5$ for an independence test. Calculate the $p$-value.

In four regions, the number of individuals who adopted a product ('Yes') or did not adopt the product ('No') was recorded. The data is presented in the following contingency table:

Calculate the $\chi^2$ statistic and the $p$-value for a test of independence at the $5\%$ significance level. State, with a reason, whether product adoption is independent of the region.

Five performance levels (Poor, Fair, Good, Very Good, Excellent) are crossed with gender, yielding a $5 \times 2$ contingency table with $\chi^2 = 13.8$.

Determine the number of degrees of freedom and the $p$-value. Use these results to draw a conclusion for the test at the $5\%$ significance level.

A $2 \times 4$ contingency table yields the following observed counts:

Calculate the $\chi^2$ statistic and test the hypothesis of independence at the $5\%$ significance level.

The following table records the preference for a new product (Like, Neutral, Dislike) by gender:

Test at the 5% level of significance whether preference for the product is independent of gender.