Each student in Mrs. Halstead’s class was assigned a random group of 10 seventh graders. They asked everyone in each group the following question: “Do you prefer vanilla or chocolate ice cream?” The results of some of the students' findings are shown below.

Question

Marcy’s group: 4 vanilla and 6 chocolate
Owen’s group: 3 vanilla and 7 chocolate
Spencer’s group: 1 vanilla and 9 chocolate
Vivian’s group: 2 vanilla and 8 chocolate
When the class combined all of the results, they found that 110 seventh graders preferred vanilla and 190 preferred chocolate. Which student’s results were closest to the class results?

Answer 1

To determine which student's results were closest to the overall class results, we'll calculate the expected proportions of vanilla and chocolate preferences based on the total class results and then compare each student's group results to these proportions.

The total preferences from the class are:

• Vanilla: 110
• Chocolate: 190

Total number of students surveyed in the class: \[ 110 + 190 = 300 \]

Proportion of students who preferred vanilla: \[ \text{Proportion of Vanilla} = \frac{110}{300} = \frac{11}{30} \approx 0.3667 \]

Proportion of students who preferred chocolate: \[ \text{Proportion of Chocolate} = \frac{190}{300} = \frac{19}{30} \approx 0.6333 \]

Now let's analyze each student's group results:

1. Marcy’s Group: 4 vanilla and 6 chocolate

   • Vanilla proportion: \( \frac{4}{10} = 0.4 \)
   • Chocolate proportion: \( \frac{6}{10} = 0.6 \)

2. Owen’s Group: 3 vanilla and 7 chocolate

   • Vanilla proportion: \( \frac{3}{10} = 0.3 \)
   • Chocolate proportion: \( \frac{7}{10} = 0.7 \)

3. Spencer’s Group: 1 vanilla and 9 chocolate

   • Vanilla proportion: \( \frac{1}{10} = 0.1 \)
   • Chocolate proportion: \( \frac{9}{10} = 0.9 \)

4. Vivian’s Group: 2 vanilla and 8 chocolate

   • Vanilla proportion: \( \frac{2}{10} = 0.2 \)
   • Chocolate proportion: \( \frac{8}{10} = 0.8 \)

Next, we will find the absolute differences between each student's results and the class results.

• Marcy’s Group:

  • Difference in vanilla: \( |0.4 - 0.3667| = 0.0333 \)
  • Difference in chocolate: \( |0.6 - 0.6333| = 0.0333 \)
  • Total difference: \( 0.0333 + 0.0333 = 0.0666 \)

• Owen’s Group:

  • Difference in vanilla: \( |0.3 - 0.3667| = 0.0667 \)
  • Difference in chocolate: \( |0.7 - 0.6333| = 0.0667 \)
  • Total difference: \( 0.0667 + 0.0667 = 0.1334 \)

• Spencer’s Group:

  • Difference in vanilla: \( |0.1 - 0.3667| = 0.2667 \)
  • Difference in chocolate: \( |0.9 - 0.6333| = 0.2667 \)
  • Total difference: \( 0.2667 + 0.2667 = 0.5334 \)

• Vivian’s Group:

  • Difference in vanilla: \( |0.2 - 0.3667| = 0.1667 \)
  • Difference in chocolate: \( |0.8 - 0.6333| = 0.1667 \)
  • Total difference: \( 0.1667 + 0.1667 = 0.3334 \)

Now we can summarize the total differences for each student group:

• Marcy: \( 0.0666 \)
• Owen: \( 0.1334 \)
• Spencer: \( 0.5334 \)
• Vivian: \( 0.3334 \)

The closest results to the class results were from Marcy with a total difference of \( 0.0666 \). Therefore, Marcy's results were the closest to the class results.