Let's analyze the problem step by step to determine which of the given statements about the joint frequencies in the table are true.
1. First, we are given the totals in each grade and the total number of students who prefer math and science.
- Ninth Grade Total: 63
- Tenth Grade Total: 26
- Eleventh Grade Total: 29
- Twelfth Grade Total: 67
- Math Total: 90
- Science Total: 95
- Overall Total: 185
2. We also have partial data for the number of students preferring math and science:
- Math:
- 10th Grade: 18
- Science:
- 9th Grade: 40
- 11th Grade: 15
- 12th Grade: 32
3. Next, let's find the missing values by using the provided information.
- To find the number of 9th graders who prefer math:
[tex]\[
\text{Math 9th} = \text{Total 9th} - \text{Science 9th}
\][/tex]
[tex]\[
\text{Math 9th} = 63 - 40 = 23
\][/tex]
- To find the number of 10th graders who prefer science:
[tex]\[
\text{Science 10th} = \text{Total 10th} - \text{Math 10th}
\][/tex]
[tex]\[
\text{Science 10th} = 26 - 18 = 8
\][/tex]
- To find the number of 11th graders who prefer math:
[tex]\[
\text{Math 11th} = \text{Total 11th} - \text{Science 11th}
\][/tex]
[tex]\[
\text{Math 11th} = 29 - 15 = 14
\][/tex]
- To find the number of 12th graders who prefer math:
[tex]\[
\text{Math 12th} = \text{Math Total} - (\text{Math 9th} + \text{Math 10th} + \text{Math 11th})
\][/tex]
[tex]\[
\text{Math 12th} = 90 - (23 + 18 + 14) = 90 - 55 = 35
\][/tex]
4. Now that we have all the missing values, let's verify each statement:
- Statement 1: "Twenty-three 9th graders and fifteen 11th graders prefer math."
- We found that 23 9th graders prefer math and 14 11th graders prefer math, not 15.
- This statement is False.
- Statement 2: "Fourteen 11th graders prefer math and eight 10th graders prefer science."
- We found that 14 11th graders prefer math and 8 10th graders prefer science.
- This statement is True.
- Statement 3: "Thirty-five 12th graders prefer math and nine 10th graders prefer science."
- We found that 35 12th graders prefer math, but 8 10th graders prefer science, not 9.
- This statement is False.
- Statement 4: "Twenty-three 9th graders and thirty-two 12th graders prefer math."
- We found that 23 9th graders prefer math and 35 12th graders prefer math, not 32.
- This statement is False.
So, the true statement is:
Fourteen 11th graders prefer math and eight 10th graders prefer science.
