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Question 1

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

\begin{array}{c|cc|c} & A & B & \text{Total}\\ \hline \text{Yes} & 20 & 30 & 50\\ \text{No} & 10 & 40 & 50\\ \hline \text{Total} & 30 & 70 & 100 \end{array}

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Question 2

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}{c|cc|c} & \text{Pass} & \text{Fail} & \text{Total} \\ \hline 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 \\ \hline \text{Total} & 350 & 150 & 500 \end{array}

Find the table of expected frequencies.

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Question 3

A survey was conducted to investigate the relationship between smoking status (Smoker, Non-smoker) and exercise frequency (Low, Moderate, High). The observed frequencies for $250$ individuals are recorded in the contingency table below:

\begin{array}{c|ccc|c} & \text{Low} & \text{Moderate} & \text{High} & \text{Total}\\ \hline \text{Smoker} & 40 & 30 & 20 & 90\\ \text{Non-smoker} & 60 & 50 & 50 & 160\\ \hline \text{Total} & 100 & 80 & 70 & 250 \end{array}

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

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Question 4

Contingency tables and expected frequencies.

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}{c|ccc|c} & A & B & O & \text{Total} \\ \hline \text{Vegetarian} & 30 & 20 & 10 & 60 \\ \text{Non-vegetarian} & 50 & 40 & 30 & 120 \\ \text{Vegan} & 20 & 10 & 40 & 70 \\ \hline \text{Total} & 100 & 70 & 80 & 250 \end{array}

Calculate the expected frequencies for each cell.

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Question 5

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

\begin{array}{c|cc|c} & \text{Buy} & \text{Don't Buy} & \text{Total}\\ \hline \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\\ \hline \text{Total} & 125 & 125 & 250 \end{array}

Find the expected-frequency table.

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Question 6

This question requires calculating the expected frequencies for a $4 \times 4$ contingency table using the formula $E = \frac{\text{row total} \times \text{column total}}{\text{grand total}}$ .

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

\begin{array}{c|cccc|c} & A & B & C & D & \text{Total} \\ \hline 1 & 10 & 20 & 30 & 40 & 100 \\ 2 & 20 & 30 & 40 & 10 & 100 \\ 3 & 30 & 10 & 20 & 40 & 100 \\ 4 & 40 & 40 & 10 & 10 & 100 \\ \hline \text{Total} & 100 & 100 & 100 & 100 & 400 \end{array}

Compute the expected-frequency table.

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Question 7

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

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

Compute the expected frequencies.

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Question 8

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

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

Find the table of expected frequencies.

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Question 9

Mathematics: Statistics and Probability

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}{c|cc|c} & \text{Healthy} & \text{Unhealthy} & \text{Total}\\ \hline \text{Under 20} & 30 & 20 & 50\\ \text{20–39} & 50 & 30 & 80\\ \text{40–59} & 40 & 40 & 80\\ \text{60+} & 20 & 30 & 50\\ \hline \text{Total} & 140 & 120 & 260 \end{array}

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

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Question 10

Questions related to Chi-squared test for independence: computing expected frequencies from a contingency table.

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

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

Compute the expected frequencies table.

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Question 11

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

\begin{array}{l|ccc|c} & \text{Low} & \text{Medium} & \text{High} & \text{Total}\\ \hline A & 15 & 25 & 10 & 50\\ B & 20 & 30 & 10 & 60\\ C & 10 & 20 & 20 & 50\\ D & 5 & 15 & 20 & 40\\ \hline \text{Total} & 50 & 90 & 60 & 200 \end{array}

Calculate the table of expected frequencies for this data.

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Question Type 4: Finding the table of expected frequencies for a chi-squared independence test Exercises