A survey was conducted with high school students in each grade to see how many prefer math or science. Some of the data are shown below.
| | 9 | 10 | 11 | 12 | Total |
| :---- | :- | :- | :- | :- | :---- |
| Math | | 18 | | | 90 |
| Science | 40 | | 15 | 32 | 95 |
| Total | 63 | 26 | 29 | 67 | 185 |
Which statement is true about the joint frequencies in this table?
A. Twenty-three 9th graders and fifteen 11th graders prefer math.
B. Fourteen 11th graders prefer math and eight 10th graders prefer science.
C. Thirty-five 12th graders prefer math and nine 10th graders prefer science.
D. Twenty-three 9th graders and thirty-two 12th graders prefer math.
Answer (2)
The true statement is that fourteen 11th graders prefer math and eight 10th graders prefer science. All calculations regarding the students' preferences were verified accurately. Therefore, based on the survey data, this option reflects the correct preferences.
;
Calculate the number of 9th graders who prefer math: 63 − 40 = 23 .
Calculate the number of 10th graders who prefer science: 26 − 18 = 8 .
Calculate the number of 11th graders who prefer math: 29 − 15 = 14 .
Calculate the number of 12th graders who prefer math: 67 − 32 = 35 .
The true statement is: 14 eleventh graders prefer math and 8 tenth graders prefer science. Fourteen 11th graders prefer math and eight 10th graders prefer science.
Explanation
Analyze the problem and data We are given a table with data about high school students in grades 9-12 and their preferences for math or science. The goal is to determine which statement about the joint frequencies in the table is true. First, we need to calculate the missing values in the table.
Calculate 9th graders who prefer math To find the number of 9th graders who prefer math, we subtract the number of 9th graders who prefer science from the total number of 9th graders: 63 − 40 = 23 . So, 23 ninth graders prefer math.
Calculate 10th graders who prefer science To find the number of 10th graders who prefer science, we subtract the number of 10th graders who prefer math from the total number of 10th graders: 26 − 18 = 8 . So, 8 tenth graders prefer science.
Calculate 11th graders who prefer math To find the number of 11th graders who prefer math, we subtract the number of 11th graders who prefer science from the total number of 11th graders: 29 − 15 = 14 . So, 14 eleventh graders prefer math.
Calculate 12th graders who prefer math To find the number of 12th graders who prefer math, we subtract the number of 12th graders who prefer science from the total number of 12th graders: 67 − 32 = 35 . So, 35 twelfth graders prefer math.
Verify the statements Now we have all the joint frequencies. Let's check the given statements:
Statement 1: Twenty-three 9th graders and fifteen 11th graders prefer math. We found that 23 ninth graders prefer math, but 14 eleventh graders prefer math, so this statement is false.
Statement 2: Fourteen 11th graders prefer math and eight 10th graders prefer science. We found that 14 eleventh graders prefer math and 8 tenth graders prefer science, so this statement is true.
Statement 3: Thirty-five 12th graders prefer math and nine 10th graders prefer science. We found that 35 twelfth graders prefer math, but 8 tenth graders prefer science, so this statement is false.
Statement 4: Twenty-three 9th graders and thirty-two 12th graders prefer math. We found that 23 ninth graders prefer math, but 35 twelfth graders prefer math, so this statement is false.
Final Answer The true statement is: Fourteen 11th graders prefer math and eight 10th graders prefer science.
Examples
Understanding joint frequencies helps in analyzing survey data to identify trends and preferences within different groups. For example, a school might use this type of analysis to determine which subjects are more popular among different grade levels, allowing them to allocate resources more effectively or tailor their curriculum to better suit student interests. This can also be applied in market research to understand consumer preferences across different demographics.
