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AP Statistics Help: Expected Counts In Two Way Tables

Review real example questions for Expected Counts In Two Way Tables in AP Statistics.

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A study examined the relationship between pet ownership (dog, cat, no pet) and allergy status (yes, no) in a random sample of 600 people. In the sample, 250 people owned a dog, 150 owned a cat, and 200 had no pet. A total of 180 people reported having allergies.

Which of the following expressions correctly calculates the expected number of people who own a cat and have allergies, under the null hypothesis of no association?

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Question 1

Which of the following expressions correctly calculates the expected number of people who own a cat and have allergies, under the null hypothesis of no association?

$\frac{(150)(180)}{600}$ (correct answer)
$\frac{(150)(250)}{600}$
$\frac{(180)(200)}{600}$
$\frac{(600)}{(150)(180)}$

Explanation: The formula for the expected count in a cell of a two-way table is (row total × column total) / grand total. Here, the row total for 'owning a cat' is 150, the column total for 'having allergies' is 180, and the grand total sample size is 600. The correct expression is $\frac{(150)(180)}{600}$ .

Question 2

Data was collected to see if there is an association between the day of the week and the number of customers at a restaurant. A random sample of 200 days was observed. The totals are: 80 weekdays and 120 weekend days. The customer traffic was categorized as Low, Medium, or High. There were 50 days with Low traffic, 90 with Medium, and 60 with High. On weekdays, there were 30 days with Low traffic.

What is the expected number of weekdays with Low customer traffic if traffic level is independent of the type of day?

$20$ (correct answer)
$30$
$40$
$50$

Explanation: The expected count is (row total × column total) / grand total. The row total for weekdays is 80. The column total for Low traffic is 50. The grand total is 200. The expected count is $(80 \times 50) / 200 = 4000 / 200 = 20$ . The value of 30 is the observed count, which is a common distractor.

Question 3

A research study produced the following two-way table of counts for two categorical variables, A and B. A total of 200 subjects were studied. For variable A, the counts are 80 for level 1 and 120 for level 2. For variable B, the counts are 100 for level X and 100 for level Y.

For a chi-square test of independence, which of the following comparisons between expected counts is correct?

The expected count for (Level 1, Level X) is equal to the expected count for (Level 2, Level X).
The expected count for (Level 1, Level X) is less than the expected count for (Level 1, Level Y).
The expected count for (Level 2, Level X) is greater than the expected count for (Level 1, Level X). (correct answer)
The expected count for (Level 1, Level Y) is greater than the expected count for (Level 2, Level Y).

Explanation: First, calculate the four expected counts using the formula (row total × column total) / grand total. The expected count for (Level 1, Level X) is $(80 \times 100) / 200 = 40$ . The expected count for (Level 1, Level Y) is $(80 \times 100) / 200 = 40$ . The expected count for (Level 2, Level X) is $(120 \times 100) / 200 = 60$ . The expected count for (Level 2, Level Y) is $(120 \times 100) / 200 = 60$ . Comparing the values as per the choices, only C is correct because the expected count for (Level 2, Level X), which is 60, is greater than the expected count for (Level 1, Level X), which is 40.

Question 4

A study was conducted to investigate whether the genre of a movie seen (Action, Comedy, Drama) is independent of the age group of the moviegoer (Child, Teen, Adult). Data was collected from a random sample of 300 moviegoers.

Under the null hypothesis that movie genre preference is independent of age group, how is the expected number of teens who prefer action movies calculated?

By multiplying the total number of teens by the total number of people who prefer action movies, then dividing by the total number of moviegoers. (correct answer)
By dividing the number of teens who were observed to prefer action movies by the total number of teens.
By multiplying the proportion of all moviegoers who are teens by the total number of action movies available.
By averaging the number of moviegoers across all combinations of age group and genre.

Explanation: The expected count for a cell under the null hypothesis of independence is calculated by the formula: (row total × column total) / grand total. In this context, this corresponds to (total number of teens × total number of people who prefer action movies) / total number of moviegoers.

Question 5

A gym tracked 500 members by whether they attend group classes and whether they renewed their membership. Assuming independence, which expression calculates the expected count for the Group Classes & Renewed cell?

$\dfrac{(200)(350)}{500}$ (correct answer)
$\dfrac{200}{500}$
$140$
$\dfrac{350}{500}$
$(200)(350)$

Explanation: To find expected counts assuming independence, we apply (row total × column total) ÷ grand total. For Group Classes & Renewed, we multiply members attending group classes (200) by members who renewed (350), then divide by all 500 members. Choice A shows this correctly: (200)(350)/500. Choice C gives 140, which is the calculated result but not the expression itself. Choices B and D show individual proportions, while E shows the product without division. The expected count formula helps us test whether the observed counts differ significantly from what independence would predict.

Question 6

A researcher classified 120 plants by whether they received fertilizer and whether they bloomed. Under the assumption of independence, which expression calculates the expected count for the No Fertilizer & Bloomed cell?

$\dfrac{(50)(70)}{120}$ (correct answer)
$\dfrac{50}{120}$
$30$
$\dfrac{70}{120}$
$(50)(70)$

Explanation: Expected counts in two-way tables use the formula (row total × column total) ÷ grand total. For No Fertilizer & Bloomed, we multiply plants without fertilizer (50) by plants that bloomed (70), then divide by all 120 plants. Choice A correctly shows (50)(70)/120. Choice C shows 30, which might be an observed count but isn't the expression. Choices B and D show marginal proportions that don't calculate expected counts, while E multiplies totals without dividing, yielding 3,500 instead of the reasonable expected count of about 29.

Question 7

A clinic categorized 180 patients by whether they received a flu shot and whether they later reported flu symptoms. Under the assumption of independence, which expression calculates the expected count for the Shot & Symptoms cell?

$\dfrac{(120)(45)}{180}$ (correct answer)
$\dfrac{45}{180}$
$\dfrac{120}{180}$
$20$
$(120)(45)$

Explanation: This problem requires calculating the expected count for a cell in a two-way table assuming independence between variables. The expected count formula is (row total × column total) ÷ grand total. For the Shot & Symptoms cell, we multiply the total who got shots (120) by the total with symptoms (45), then divide by all 180 patients. Choice A correctly represents this: (120)(45)/180. Choice D shows 20, which might be an observed count, while B and C show individual proportions. Choice E multiplies the totals but forgets the crucial step of dividing by the grand total.

Question 8

A study categorized 90 commuters by whether they bike to work and whether their commute is under 5 miles or 5 miles and over. Assuming independence, which expression calculates the expected count for the Bike & Under 5 miles cell?

$\dfrac{50}{90}$
$\dfrac{(36)(50)}{90}$ (correct answer)
$\dfrac{36}{90}$
$18$
$(36)(50)$

Explanation: Expected counts under independence use the formula $(row\ total \times column\ total) \div grand\ total$ . For Bike & Under 5 miles, we multiply commuters who bike (36) by those with commutes under 5 miles (50), then divide by all 90 commuters. Choice B shows this correctly: $(36)(50)/90$ . Choice D gives 18, which might be an observed count rather than the expression. Choices A and C show individual proportions, while E shows only the product. Understanding this formula is essential for testing whether categorical variables are associated or independent.

Question 9

A school surveyed 200 students about whether they participate in a sport and whether they prefer morning or afternoon classes. Under the assumption that sport participation and class-time preference are independent, which expression calculates the expected count for the Sport & Morning cell?

$\dfrac{(120)(110)}{200}$ (correct answer)
$\dfrac{120}{200}$
$\dfrac{110}{200}$
$70$
$(120)(110)$

Explanation: This question tests your ability to calculate expected counts in a two-way table under the assumption of independence. The formula for expected count is (row total × column total) ÷ grand total. Since we need the expected count for Sport & Morning, we multiply the total number of students who play sports (120) by the total number who prefer morning classes (110), then divide by the grand total of 200 students. Choice A correctly shows this formula: (120)(110)/200. Choice D gives 70, which might be the actual count but not the expression for calculating it. The other choices show only partial calculations or incorrect formulas.

Question 10

A university surveyed 300 students about whether they live on campus and whether they own a car. Assuming these variables are independent, which expression calculates the expected count for the On Campus & Owns Car cell?

$\dfrac{(180)(120)}{300}$ (correct answer)
$\dfrac{180}{300}$
$\dfrac{120}{300}$
$80$
$(180)(120)$

Explanation: Expected counts in two-way tables are calculated using (row total × column total) ÷ grand total when assuming independence. For the On Campus & Owns Car cell, we multiply students living on campus (180) by students owning cars (120), then divide by the total 300 students. Choice A shows this correctly: (180)(120)/300. Choice D gives 80, which could be the actual count but isn't the expression. Choices B and C show individual proportions, while E shows the product without dividing by the grand total, a common error.