IB Mathematics Applications & Interpretation Question Type 4: Finding the Table of Expected Frequencies for a Chi-squared Independence Test Exercises

Consider the contingency table below. Find the table of expected frequencies for a chi-squared test of independence.

$$\begin{array}{cccc}& A& B& \text{Total}\\ \text{Yes}& 20& 30& 50\\ \text{No}& 10& 40& 50\\ \text{Total}& 30& 70& 100\end{array}$$

An educational study cross-tabulates five teaching methods ($M_1$ to $M_5$) by student outcome (Pass, Fail). The following table shows the observed frequencies:

$$\begin{array}{cccc}& \text{Pass}& \text{Fail}& \text{Total}\\ M_1& 80& 20& 100\\ M_2& 70& 30& 100\\ M_3& 60& 40& 100\\ M_4& 90& 10& 100\\ M_5& 50& 50& 100\\ \text{Total}& 350& 150& 500\end{array}$$

Find the table of expected frequencies.

Determine the expected frequencies for all cells in the contingency table.

A group of individuals were surveyed about their blood type and their dietary preferences. The results are shown in the following $3\times 3$ contingency table:

$$\begin{array}{ccccc}& A& B& O& \text{Total}\\ \text{Vegetarian}& 30& 20& 10& 60\\ \text{Non-vegetarian}& 50& 40& 30& 120\\ \text{Vegan}& 20& 10& 40& 70\\ \text{Total}& 100& 70& 80& 250\end{array}$$

Calculate the expected frequencies for each cell.

In a marketing study, 5 age brackets and 2 purchase decisions (Buy, Don't Buy) produced the following contingency table:

$$\begin{array}{cccc}& \text{Buy}& \text{Don’t Buy}& \text{Total}\\ \text{<25}& 30& 20& 50\\ 25\text{–}34& 40& 10& 50\\ 35\text{–}44& 20& 30& 50\\ 45\text{–}54& 10& 40& 50\\ 55+& 25& 25& 50\\ \text{Total}& 125& 125& 250\end{array}$$

Find the expected-frequency table.

A $4\times 4$ contingency table records satisfaction levels ($1$–$4$) and service types ($A$–$D$):

$$\begin{array}{cccccc}& A& B& C& D& \text{Total}\\ 1& 10& 20& 30& 40& 100\\ 2& 20& 30& 40& 10& 100\\ 3& 30& 10& 20& 40& 100\\ 4& 40& 40& 10& 10& 100\\ \text{Total}& 100& 100& 100& 100& 400\end{array}$$

Compute the expected-frequency table.

A $2\times 4$ table shows device usage (Desktop, Mobile) across four regions (North, South, East, West):

$$\begin{array}{cccccc}& \text{North}& \text{South}& \text{East}& \text{West}& \text{Total}\\ \text{Desktop}& 40& 30& 20& 10& 100\\ \text{Mobile}& 60& 70& 80& 90& 300\\ \text{Total}& 100& 100& 100& 100& 400\end{array}$$

Compute the expected frequencies.

A teacher records student performance (Pass, Fail) in three different courses (Math, Science, English). Observed counts are:

$$\begin{array}{ccccc}& \text{Math}& \text{Science}& \text{English}& \text{Total}\\ \text{Pass}& 80& 70& 90& 240\\ \text{Fail}& 20& 30& 10& 60\\ \text{Total}& 100& 100& 100& 300\end{array}$$

Find the table of expected frequencies.

In a study of four age groups (Under 20, 20–39, 40–59, 60+) and two health statuses (Healthy, Unhealthy), the observed data are given in the following contingency table:

$$\begin{array}{cccc}& \text{Healthy}& \text{Unhealthy}& \text{Total}\\ \text{Under 20}& 30& 20& 50\\ \text{20–39}& 50& 30& 80\\ \text{40–59}& 40& 40& 80\\ \text{60+}& 20& 30& 50\\ \text{Total}& 140& 120& 260\end{array}$$

Compute the expected frequencies table. Give your answers to three significant figures.

A researcher surveys preferences among three flavors (Vanilla, Chocolate, Strawberry) by gender. The observed counts are:

$$\begin{array}{ccccc}& \text{Vanilla}& \text{Chocolate}& \text{Strawberry}& \text{Total}\\ \text{Male}& 25& 30& 15& 70\\ \text{Female}& 20& 25& 35& 80\\ \text{Total}& 45& 55& 50& 150\end{array}$$

Compute the expected frequencies table.

A survey of four departments ($A,B,C,D$) and three satisfaction levels (Low, Medium, High) yields the following contingency table:

$$\begin{array}{lcccc}& \text{Low}& \text{Medium}& \text{High}& \text{Total}\\ A& 15& 25& 10& 50\\ B& 20& 30& 10& 60\\ C& 10& 20& 20& 50\\ D& 5& 15& 20& 40\\ \text{Total}& 50& 90& 60& 200\end{array}$$

Calculate the table of expected frequencies for this data.