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 class results, we'll compare each group's results to the overall results of 110 vanilla and 190 chocolate preferences.

First, let’s calculate the total number of students surveyed:

• Total students = 110 (vanilla) + 190 (chocolate) = 300 students.

Next, we calculate the proportion of students who preferred vanilla and chocolate based on the class results:

• Proportion preferring vanilla = 110 / 300 = 0.3667 (approximately)
• Proportion preferring chocolate = 190 / 300 = 0.6333 (approximately)

Now we will evaluate each student's group results:

1. Marcy’s group:

   • Vanilla: 4 out of 10 (0.4 or 40%)
   • Chocolate: 6 out of 10 (0.6 or 60%)

2. Owen’s group:

   • Vanilla: 3 out of 10 (0.3 or 30%)
   • Chocolate: 7 out of 10 (0.7 or 70%)

3. Spencer’s group:

   • Vanilla: 1 out of 10 (0.1 or 10%)
   • Chocolate: 9 out of 10 (0.9 or 90%)

4. Vivian’s group:

   • Vanilla: 2 out of 10 (0.2 or 20%)
   • Chocolate: 8 out of 10 (0.8 or 80%)

Now, let’s calculate how close each group’s proportions are to the overall proportions:

1. Marcy’s group:

   • (Vanilla) | Difference = | 0.3667 - 0.4 | = 0.0333
   • (Chocolate) | Difference = | 0.6333 - 0.6 | = 0.0333
   • Total difference = 0.0333 + 0.0333 = 0.0666

2. Owen’s group:

   • (Vanilla) | Difference = | 0.3667 - 0.3 | = 0.0667
   • (Chocolate) | Difference = | 0.6333 - 0.7 | = 0.0667
   • Total difference = 0.0667 + 0.0667 = 0.1334

3. Spencer’s group:

   • (Vanilla) | Difference = | 0.3667 - 0.1 | = 0.2667
   • (Chocolate) | Difference = | 0.6333 - 0.9 | = 0.2667
   • Total difference = 0.2667 + 0.2667 = 0.5334

4. Vivian’s group:

   • (Vanilla) | Difference = | 0.3667 - 0.2 | = 0.1667
   • (Chocolate) | Difference = | 0.6333 - 0.8 | = 0.1667
   • Total difference = 0.1667 + 0.1667 = 0.3334

Based on the total differences calculated:

• Marcy's group has the smallest total difference (0.0666).
• Owen's group has a total difference of 0.1334.
• Spencer's group has a total difference of 0.5334.
• Vivian's group has a total difference of 0.3334.

Thus, Marcy's results were closest to the class results.