Answered

Students at a virtual school are allowed to sign up for one math class each year. The numbers of students signing up for various math classes for the next school year are given in the following table:




Grade		GeometryAlgebra IIPre-CalculusAP StatisticsTotal

10th		20.7\%10.3\%3.4\%0.7\%35.2\%

11th		6.7\%13.8\%10.3\%2.8\%33.8\%

12th		1.4\%6.7\%13.8\%9\%31\%

Total		29\%31\%27.6\%12.4\%1






A student calculated the joint relative frequency of 10th grade students in geometry as being 71.4%. What did the student actually calculate, and what is the correct answer?


AThe student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 20.7%.
BThe student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 29%.
CThe student calculated the marginal relative frequency for 10th grade students in geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 20.7%.
DThe student calculated the marginal relative frequency for 10th grade students in geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 20%.

Verified Answer :

  • The student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry.
  • The calculation is: \(P(\text{10th} | \text{Geometry}) = \frac{P(\text{10th} \cap \text{Geometry})}{P(\text{Geometry})} = \frac{20.7\%}{29\%} \approx 71.4\%\).
  • The correct joint relative frequency of 10th grade students in geometry is found directly from the table intersection.
  • The correct answer is: \(\boxed{\text{The student calculated the conditional relative frequency for students who are in 10th grade, given that they are enrolled in Geometry. The correct value of the joint relative frequency of 10th grade students in geometry is 20.7\%.}}\)

Explanation

  1. Analyze the data and the student's claim

The problem provides a table of joint relative frequencies for students by grade level and math class enrollment. We are looking at the specific cell for 10th-grade students in Geometry, which has a joint relative frequency of \(20.7\%\). A student claimed this value is \(71.4\%\). We need to determine what calculation resulted in \(71.4\%\) and verify the correct joint relative frequency.

  1. Evaluate conditional probabilities

We have two types of conditional probabilities involving 10th-grade students and Geometry:

  1. The probability that a student is in 10th grade, given that they are in Geometry: \(P(\text{10th} | \text{Geometry}) = \frac{P(\text{10th} \cap \text{Geometry})}{P(\text{Geometry})}\).
  2. The probability that a student is in Geometry, given that they are in 10th grade: \(P(\text{Geometry} | \text{10th}) = \frac{P(\text{10th} \cap \text{Geometry})}{P(\text{10th})}\).

Using the values from the table:
\(P(\text{10th} \cap \text{Geometry}) = 20.7\%\)
\(P(\text{Geometry}) = 29\%\)
\(P(\text{10th}) = 35.2\%\)

Calculating the first conditional probability:
\(\frac{20.7\%}{29\%} \approx 0.7138\), which is approximately \(71.4\%\).

Calculating the second conditional probability:
\(\frac{20.7\%}{35.2\%} \approx 0.5881\), which is approximately \(58.8\%\).

  1. Conclusion

The calculation \(\frac{20.7\%}{29\%} \approx 71.4\%\) shows that the student actually calculated the conditional relative frequency of being in 10th grade, given that the student is enrolled in Geometry. The correct joint relative frequency for 10th-grade students in geometry is the value located at the intersection of the "10th" row and "Geometry" column in the provided table, which is \(20.7\%\).

Examples

In school administration, these calculations are used to understand enrollment trends. For example, knowing that \(P(\text{10th} | \text{Geometry}) \approx 71.4\%\) helps administrators see that the vast majority of students currently enrolled in Geometry are 10th graders, which might influence staffing or curriculum planning for that specific course. Joint relative frequencies, like \(20.7\%\), provide the foundational data for all such insights, showing the portion of the entire student population that falls into both categories.

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