# Support for a Longer School Day?

Alignments to Content Standards: S-ID.B.5

Each student in a random sample of students at a local high school was categorized according to gender (male or female) and whether they supported a proposal to increase the length of the school day by 30 minutes (oppose, support, no opinion). The following table summarizes the data for this sample.

 Opinion on Proposal to Increase Length of School Day Oppose Support No Opinion Total Gender Male 50 40 20 110 Female 40 40 10 90 Total 90 80 30 200

1. What proportion of the students in this sample are male?

2. What proportion of the students in this sample support the proposal?

3. What proportion of the males in this sample support the proposal?

4. What proportion of the students in this sample who support this proposal are male?

5. Interpret the following joint relative frequency in the context of this problem: 10/200.

6. Interpret the following marginal relative frequency in the context of this problem: 30/200.

7. Interpret the following conditional frequency in the context of this problem: 50/110.

8. Interpret the following conditional frequency in the context of this problem: 20/110.

9. Interpret the following conditional frequency in the context of this problem: 20/30.

## IM Commentary

The purpose of this task is to provide students with an opportunity to calculate joint, marginal and relative frequencies using data in a two-way table (S.ID.5). Students are also asked to interpret joint, marginal and conditional relative frequencies in context. In the discussion of this task, focus on the difference between the three types of relative frequencies and their interpretations. If students struggle with parts (e) through (i), have them look at the table to see where the numerator and denominator of the given relative frequency appear in the data table. In particular, have them note whether the denominator is the grand total for the table (which would mean that the relative frequency is either a joint or marginal relative frequency) or a row or column total (which would mean that the given relative frequency is a conditional relative frequency).

## Solution

1. ${number \ of \ males \over total \ number \ of \ students} = {110 \over 200} = 0.55$

2. ${number \ who \ support \ proposal \over total \ number \ of \ students} = {80 \over 200} = 0.40$

3. ${number \ of \ males \ who \ support \ the \ proposal \over total \ number \ of \ males} = {40 \over 110} = 0.39$

4. ${number \ of \ males \ who \ support \ the \ proposal \over total \ number \ who \ support \ the \ proposal} = {40 \over 80} = 0.50$

5. This is the proportion of students in the sample who are female and who have no opinion on the proposal.

6. This is the proportion of students in the sample who have no opinion on the proposal.

7. This is the proportion of males in the sample who oppose the proposal.

8. This is the proportion of males in the sample who have no opinion on the proposal.

9. This is the proportion of students in the sample who have no opinion on the proposal who are male.