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Question 1

Skill question

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}

Question 2

Skill question

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 expected‐frequencies table.

Question 3

Skill question

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.

Question 4

Skill question

A 2×4 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.

Question 5

Skill question

The following table shows smoking status (Smoker, Non-smoker) by exercise frequency (Low, Moderate, High). Determine the expected frequencies.

\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}

Question 6

Skill question

In a marketing study, 5 age brackets and 2 purchase decisions (Buy, Don’t Buy) produced:

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

Find the expected‐frequency table.

Question 7

Skill question

An educational study cross‐tabulates five teaching methods (M1–M5) by student outcome (Pass, Fail):

\begin{array}{c|cc|c} & \text{Pass} & \text{Fail} & \text{Total}\\ \hline M1 & 80 & 20 & 100\\ M2 & 70 & 30 & 100\\ M3 & 60 & 40 & 100\\ M4 & 90 & 10 & 100\\ M5 & 50 & 50 & 100\\ \hline \text{Total} & 350 & 150 & 500 \end{array}

Find the table of expected frequencies.

Question 8

Skill question

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

\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.

Question 9

Skill question

Survey of four departments (A, B, C, D) and three satisfaction levels (Low, Medium, High) yields:

\begin{array}{c|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}

Find the expected‐frequency table.

Question 10

Skill question

A 3×3 table shows blood type (A, B, O) by diet (Vegetarian, Non-vegetarian, Vegan):

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

Compute expected frequencies.

Question 11

Skill question

A 4×4 contingency table records satisfaction (1–4) and service type (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.

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