Two-way tables in 8th grade

The answer is 25% because 4 out of the 16 students who prefer swimming are in the younger age group.

1. 1Look at the Swimming column and the Younger row. The cell where they meet is 4.
2. 2For a column relative frequency, the whole is the total of that column. The total at the bottom of the Swimming column is 16.
3. 3Divide the Younger count by the Swimming column total.

Common mistake. A common mistake is dividing by the Younger group row total of 8. That gives a row relative frequency, but this question asks for a column relative frequency, so the denominator must be 16.

The idea underneath. Column relative frequency always uses the column total as the whole, so you ask what share of that column belongs to the named row.

Maya's setup is correct. The denominator 30 is the pet-owner group total, the right denominator for this question, and the fraction $\frac{18}{30}$ correctly represents that conditional fraction.

1. 1Read the group named in the question: it points to students who are in the 'Have a pet' row. That group determines the denominator.
2. 2In the table, the total number of students in the 'Have a pet' group is 30 — read this from the 'Have a pet' row total.
3. 3The numerator 18 is correct because 18 students are in both the 'Have a pet' row and the 'Walk to school' column — this is the intersection cell.
4. 4Maya's denominator 30 is exactly the 'Have a pet' group total from step 2 — it already matches the conditioned-group total. The displayed fraction $\frac{18}{30}$ is the correct setup, so the best description is 'No mistake; the setup is correct.'

Common mistake. A common mistake is assuming the setup must contain an error without checking the denominator. Maya's denominator 30 is exactly the pet-owner group total, not the school-walker group total (33), the non-pet-owner group total (40), or the grand total (70).

The idea underneath. In a conditional proportion, the denominator is the group named after 'of' or 'among,' and the numerator is the part of that same group that also fits the second category.

The correct table shows Soccer 3 and 4 with row total 7, Basketball 2 and 1 with row total 3, column totals 5 and 5, and grand total 10.

1. 1Each row in the survey table is 1 student. Count the students in each pair of categories.
2. 2The joint counts are: Soccer and Yes = 3, Soccer and No = 4, Basketball and Yes = 2, and Basketball and No = 1.
3. 3Now add the row totals, column totals, and grand total.
4. 4The matching table must show Soccer: 3 yes, 4 no, total 7; Basketball: 2 yes, 1 no, total 3; column totals 5 and 5; and grand total 10.

Common mistake. A common mistake is getting the grand total of 10 correct but swapping the Yes and No counts in the Soccer row. Count each category pair first, then fill in the row and column totals.

The idea underneath. In a two-way table, each subject belongs in exactly 1 interior cell, and every total comes from adding those interior cell counts.

87.5% and 55%; students who do not play a sport are less likely to have a sibling.

1. 1Read the 'Plays a sport' row: 35 students have a sibling out of 40 total. Read the 'Does not play a sport' row: 11 students have a sibling out of 20 total. Use the row totals (40 and 20) as the denominators because you want the percent of each group that has a sibling, not the share of the whole survey.
2. 2Compute the percent who have a sibling among students who play a sport. Dividing 35 by 40 gives 0.875, which equals exactly 87.5%. Enter 87.5 in the first input (input-rate-with).
3. 3Compute the percent who have a sibling among students who do not play a sport. Dividing 11 by 20 gives 0.55, which equals 55%. Enter 55 in the second input (input-rate-without).
4. 4Compare the two rates. Because 55% < 87.5%, students who do not play a sport are less likely to have a sibling than students who play a sport. Choose 'less likely' from the dropdown.

Common mistake. A common mistake is comparing the raw counts 35 and 11, or dividing both by the grand total 60. The groups are different sizes, so the fair comparison is $\frac{35}{40}$ versus $\frac{11}{20}$.

The idea underneath. To describe association in a two-way table, compare conditional rates within each group, not raw counts from groups of different sizes.