Expected Counts in Two-Way Tables
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AP Statistics › Expected Counts in Two-Way Tables
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?
$(200)(350)$
$140$
$\dfrac{200}{500}$
$\dfrac{(200)(350)}{500}$
$\dfrac{350}{500}$
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.
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?
$\dfrac{(50)(70)}{120}$
$\dfrac{70}{120}$
$30$
$\dfrac{50}{120}$
$(50)(70)$
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.
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?
$\dfrac{45}{180}$
$\dfrac{120}{180}$
$\dfrac{(120)(45)}{180}$
$(120)(45)$
$20$
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.
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?
$\dfrac{36}{90}$
$\dfrac{(36)(50)}{90}$
$18$
$\dfrac{50}{90}$
$(36)(50)$
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.
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?
$70$
$\dfrac{110}{200}$
$\dfrac{(120)(110)}{200}$
$(120)(110)$
$\dfrac{120}{200}$
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.
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?
$\dfrac{120}{300}$
$\dfrac{(180)(120)}{300}$
$80$
$\dfrac{180}{300}$
$(180)(120)$
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.
A company recorded whether 150 employees work remotely and whether they are in the Sales or Engineering department. Assuming the two variables are independent, which expression calculates the expected count for the Remote & Sales cell?
$\dfrac{60}{150}$
$\dfrac{(90)(60)}{150}$
$30$
$(90)(60)$
$\dfrac{90}{150}$
Explanation
To find expected counts in two-way tables, we use the formula (row total × column total) ÷ grand total. For the Remote & Sales cell, we need to multiply the total number of remote workers (90) by the total number in Sales (60), then divide by the grand total of 150 employees. Choice B correctly shows this: (90)(60)/150. Choice C shows 30, which is likely the result of the calculation but not the expression itself. Choices A and D show individual proportions rather than the complete formula, while E forgets to divide by the grand total.
A bookstore recorded 160 purchases by whether the customer used a coupon and whether the purchase was fiction or nonfiction. If coupon use and genre are independent, which expression calculates the expected count for the Coupon & Fiction cell?
$\dfrac{(40)(100)}{160}$
$\dfrac{100}{160}$
$(40)(100)$
$20$
$\dfrac{40}{160}$
Explanation
This problem requires the expected count formula: (row total × column total) ÷ grand total. For Coupon & Fiction, we multiply customers using coupons (40) by fiction purchases (100), then divide by all 160 purchases. Choice A correctly represents this: (40)(100)/160. Choice D shows 20, which might be an actual count but not the expression. Choices B and C show marginal proportions that don't give expected counts, while E forgets to divide by the grand total. Expected counts are crucial for chi-square tests of independence.
A poll of 400 adults recorded whether they support a policy and whether they are under 30 or 30 and over. Assuming independence, which expression calculates the expected count for the Under 30 & Support cell?
$\dfrac{240}{400}$
$(150)(240)$
$\dfrac{160}{400}$
$90$
$\dfrac{(150)(240)}{400}$
Explanation
This question tests the expected count formula: (row total × column total) ÷ grand total. For Under 30 & Support, we multiply those under 30 (150) by those who support (240), then divide by the total 400 adults. Choice B shows this correctly: (150)(240)/400. Choice C gives 90, which is the calculated result (36,000/400 = 90) but not the expression. Choices A and D show individual proportions, while E shows the product without division. Remember that expected counts tell us what we'd expect if the variables were truly independent.
A movie theater recorded 250 customers by whether they bought popcorn and whether they attended a matinee or evening show. If popcorn purchase and show time are independent, which expression calculates the expected count for the Popcorn & Evening cell?
$\dfrac{100}{250}$
$(100)(150)$
$\dfrac{150}{250}$
$\dfrac{(100)(150)}{250}$
$60$
Explanation
To calculate expected counts under independence, we use (row total × column total) ÷ grand total. For Popcorn & Evening, we need the total who bought popcorn (100) times the total at evening shows (150), divided by all 250 customers. Choice B correctly shows (100)(150)/250. Choice C shows 60, likely the calculated result rather than the expression. Choices A and D show individual proportions that don't give expected counts, while E forgets to divide by the grand total, which would give an impossibly large expected count.