Answered

Out of 106 total sophomores, there were forty-eight boys who received either an A, B, or C on their first math test. Out of the twenty-eight total A's, sixteen girls received A's. Out of the total fifty-four B's, thirty girls received B's. Twelve boys received a C out of the twenty-four total C's on the test. Which two-way table represents this information?



Option 1:


ABCTotal

Boys		448424106

Girls		28541258

Total		16301248






Option 2:


ABCTotal

Boys		12241248

Girls		16301258

Total		285424106






Option 3:


ABCTotal

Boys		16301258

Girls		12241248

Total		285424106

Verified Answer :

  • Calculate the number of boys and girls for each grade: \(Boys_A = 12, Boys_B = 24, Boys_C = 12, Girls_A = 16, Girls_B = 30, Girls_C = 12\).
  • Verify the total count for boys (\(48\)) and girls (\(58\)).
  • Select the second table, which accurately reflects these values.
  • Final answer: \(\boxed{\text{The second table}}\).

Explanation

  1. Analyze and calculate the distribution of grades by gender

To determine the correct two-way table, we need to organize the data regarding the grades (A, B, C) by gender (Boys, Girls).

The provided data is as follows:

  • Total students: \(106\).
  • Total grade A's: \(28\), with \(16\) of these received by girls.
  • Total grade B's: \(54\), with \(30\) of these received by girls.
  • Total grade C's: \(24\), with \(12\) of these received by boys.
  • Total boys who received an A, B, or C: \(48\).

First, let's find the number of boys and girls for each grade:

  • For grade A: \(Boys_A = Total_A - Girls_A = 28 - 16 = 12\).
  • For grade B: \(Boys_B = Total_B - Girls_B = 54 - 30 = 24\).
  • For grade C: \(Girls_C = Total_C - Boys_C = 24 - 12 = 12\).

Now, we can verify the total count of boys and girls:

  • Total Boys: \(Boys_A + Boys_B + Boys_C = 12 + 24 + 12 = 48\).
  • Total Girls: \(Girls_A + Girls_B + Girls_C = 16 + 30 + 12 = 58\).
  • Check the sum: \(48 + 58 = 106\), which matches the total number of students.
  1. Compare the results with the provided tables and conclude

Now we compare our derived values with the three given tables:

  • Rows:
  • Boys: A = \(12\), B = \(24\), C = \(12\), Total = \(48\)
  • Girls: A = \(16\), B = \(30\), C = \(12\), Total = \(58\)
  • Columns:
  • Total A: \(12 + 16 = 28\)
  • Total B: \(24 + 30 = 54\)
  • Total C: \(12 + 12 = 24\)
  • Overall Total: \(28 + 54 + 24 = 106\)

Comparing this to the provided tables, the second table matches these values perfectly:

ABCTotal \) Boys\(12\)\(24\)\(12\)\(48\) \) Girls\(16\)\(30\)\(12\)\(58\) \) Total\(28\)\(54\)\(24\)\(106\) \)

Examples

Understanding how to organize categorical data into a two-way table is crucial in real-world scenarios such as reporting student performance or analyzing survey results. For example, if a company wants to compare employee satisfaction levels across different departments, they can create a two-way table with departments as rows and satisfaction ratings (e.g., High, Medium, Low) as columns to quickly identify trends. In our case, the two-way table effectively displays the distribution of math grades by gender, allowing for clear comparison and interpretation of the data.

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