IB Mathematics Applications & Interpretation 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:

$$\begin{array}{lcc}& \text{Disease Present}& \text{Disease Absent}\\ \text{Smoker}& 40& 60\\ \text{Non-Smoker}& 30& 70\end{array}$$

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:

$$\begin{array}{ccc}& \text{Recovered}& \text{Not Recovered}\\ \text{Treatment A}& 50& 30\\ \text{Treatment B}& 20& 40\end{array}$$

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:

$$\begin{array}{ccccc}& \text{Category 1}& \text{Category 2}& \text{Category 3}& \text{Category 4}\\ \text{Group 1}& 10& 20& 30& 40\\ \text{Group 2}& 30& 25& 20& 15\end{array}$$

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.