Two-Way Tables — Challenge
Two yes-or-no questions, four boxes, and a set of totals that must agree. Then turn counts into percentages.
What these problems cover
These questions use 3 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Association — A pattern linking the two variables.
- Two-way table — A table of counts sorted by two categorical variables at once.
Problem 1
In a survey of 200 people, 120 own a bike and 70 of those also own a car. Altogether 130 own a car. How many own neither?
- 20
- 10
- 50
- 60
- 80
Worked answer
20
Bike and car: 70. Bike only: 120 - 70 = 50.
Car only: 130 - 70 = 60.
Total with at least one: 70 + 50 + 60 = 180.
Neither: 200 - 180 = 20.
Problem 2
Of 80 students, 45 play a sport and 30 play an instrument; 12 do both. What percent of the sport players also play an instrument?
- About 27%
- 12%
- 40%
- 15%
- About 37%
Worked answer
About 27%
"Of the sport players" means divide by 45.
1245 = 0.26.
About 27%.
Dividing by 80 or by 30 would answer a different question.
Problem 3
A two-way table shows 30 of 50 boys and 36 of 90 girls chose the science elective. Which is true?
- A larger share of boys chose it, even though more girls are in the school
- More boys chose it than girls
- The same share of each chose it
- A larger share of girls chose it
- No comparison is possible without the totals
Worked answer
A larger share of boys chose it, even though more girls are in the school
Boys: 3050 = 60%.
Girls: 3690 = 40%.
So a larger share of boys chose it.
Raw counts (30 against 36) point the other way, which is the trap.
Problem 4
In a two-way table, a row total is 48 and the grand total is 150. One cell in that row is 18. Which statement is certain?
- 18 is 37.5% of that row
- 18 is 37.5% of the grand total
- The other cell in the row is 132
- The column total is 48
- 18 is 12% of the grand total
Worked answer
18 is 37.5% of that row
Row percentage divides by the row total: 1848 = 0.375.
So 37.5% of that row.
18150 = 12% is the table percentage, a different figure — and the option pairing it with 37.5% is the row one.
Problem 5
Two variables in a two-way table show no association. Which pattern would you expect?
- The row percentages are about the same in every row
- Every cell holds the same count
- The row totals are all equal
- The grand total is even
- One cell is zero
Worked answer
The row percentages are about the same in every row
No association means knowing the row tells you nothing about the column.
So the split across columns looks the same in each row.
Equal counts would need equal group sizes too, which is a stronger claim.
Problem 6
A table records 60 people by two yes/no questions. The four inner cells are x, 14, 21 and 10. Find x.
- 15
- 16
- 14
- 25
- 11
Worked answer
15
Every person falls in exactly one inner cell.
So the four cells add to the grand total.
x + 14 + 21 + 10 = 60.
x = 60 - 45 = 15.