AP Statistics Two Categorical Variables and Two-Way Tables Worksheet Solutions (PDF)

The explanatory variable is the one that may help explain the other.

(a) Explanatory: type of breakfast; response: alertness. (b) Explanatory: grade level; response: buying lunch.

No bike: sophomores \(40-24=16\), juniors \(60-30=30\). Row totals: \(24+30=54\) and \(16+30=46\).

54 own a bike and 46 do not.

Reads daily: \(\frac{70}{200}=35\%\). Grade 10: \(\frac{70}{200}=35\%\).

35% and 35%.

The cell is 50 out of the grand total 200: \(\frac{50}{200}=25\%\).

25%.

Grade 9: \(\frac{40}{70}\approx 57.1\%\). Grade 11: \(\frac{10}{60}\approx 16.7\%\).

About 57.1% and 16.7%.

The row total is 70.

1. \(\frac{40}{70}\approx 57.1\%\)
2. \(\frac{20}{70}\approx 28.6\%\)
3. \(\frac{10}{70}\approx 14.3\%\)

57.1%, 28.6%, 14.3% (they add to 100%).

Percent who read daily within each grade: 57%, 29%, 17% (rounded).

The percent who read daily falls steadily from Grade 9 to Grade 11, so reading habit and grade are associated: younger students read daily more often.

Allowed: \(0.78\times 200=156\) passed. Banned: \(0.84\times 200=168\) passed.

156 and 168. The pass percents differ (78% compared with 84%), so there is an association, but not necessarily causation.

Smokers who cough: \(\frac{60}{150}=40\%\). Coughers who smoke: \(\frac{60}{100}=60\%\) (100 coughers in all).

40% of smokers cough; 60% of coughers smoke. The two conditionals are different questions.

Girls: 200; in band \(0.30\times 200=60\). Boys: 200; in band \(0.20\times 200=40\).

Band: 60 girls, 40 boys; not in band: 140 girls, 160 boys.

Easy: X \(\frac{72}{80}=90\%\), Y \(\frac{10}{10}=100\%\). Hard: X \(\frac{2}{20}=10\%\), Y \(\frac{12}{90}\approx 13.3\%\).

Overall: X \(\frac{74}{100}=74\%\), Y \(\frac{22}{100}=22\%\).

Y is better in each course, but X is better overall.

Program X enrolled mostly students in the easy course (80 of 100), where passing is common. Program Y enrolled mostly students in the hard course (90 of 100).

Course difficulty is a lurking variable; the combined data hide it. Compare within courses.

The conditional percents of preferring online classes are equal in both groups.

No association: knowing a student’s sex does not change the percent who prefer online classes.

The study shows an association in observational data.

No. Students who eat breakfast may differ in sleep, family support, or other lurking variables. Only a randomized experiment can support a cause-and-effect claim.

Cat owners: \(\frac{48}{60}=80\%\). Dog owners: \(\frac{32}{140}\approx 22.9\%\).

80% compared with about 22.9%. The difference is large, so type of pet owned and opinion of the ad are associated.

Use comparison words and the context.

Cat owners were much more likely to like the ad than dog owners (80% compared with 22.9%). Among dog owners, 77.1% disliked the ad.

Females who prefer coffee: \(140-63=77\). Total coffee: \(250-90=160\). Males who prefer coffee: \(160-77=83\).

83 males prefer coffee.

Overall tea: \(\frac{90}{250}=36\%\). Expected: \(0.36\times 140=50.4\).

Expected 50.4 females prefer tea if there were no association; the observed 63 is higher.

Rural: \(\frac{540}{1500}=36\%\). Non-rural: teens \(4000-1500=2500\), vaping \(1200-540=660\), so \(\frac{660}{2500}=26.4\%\).

36% of rural teens and 26.4% of non-rural teens vape.

Bus riders: \(0.35\times 600=210\). Grade 9 riders: \(0.40\times 210=84\).

Percent of all students: \(\frac{84}{600}=14\%\).

84 students, 14% of all students.