To determine the sample proportion of students who took fewer than the mean number of electives, we need to follow these steps:
1. Calculate the Mean Number of Electives:
The first thing we should do is calculate the mean (average) number of electives taken by the students. The mean is calculated by summing all the values and then dividing by the number of values.
Here are the number of electives taken by the 19 students:
[tex]\[ 6, 6, 8, 7, 7, 7, 8, 9, 10, 8, 7, 6, 9, 6, 8, 7, 9, 7, 10 \][/tex]
Sum of electives: [tex]\(6 + 6 + 8 + 7 + 7 + 7 + 8 + 9 + 10 + 8 + 7 + 6 + 9 + 6 + 8 + 7 + 9 + 7 + 10 = 145\)[/tex]
Number of students: [tex]\(19\)[/tex]
Mean number of electives:
[tex]\[
\text{Mean} = \frac{\text{Sum of all electives}}{\text{Number of students}} = \frac{145}{19} \approx 7.631578947368421
\][/tex]
2. Count the Students Who Took Fewer Than the Mean Number of Electives:
We compare each student's number of electives with the calculated mean. We count how many students took fewer electives than the mean.
Students with electives fewer than the mean (7.63):
[tex]\[
6, 6, 7, 7, 7, 6, 6, 7, 7
\][/tex]
Number of students in this category: [tex]\(10\)[/tex]
3. Calculate the Sample Proportion:
The sample proportion is the fraction of students who took fewer than the mean number of electives. This is calculated by dividing the number of students who took fewer electives by the total number of students.
Number of students who took fewer than the mean: [tex]\(10\)[/tex]
Total number of students: [tex]\(19\)[/tex]
Sample proportion:
[tex]\[
\text{Sample proportion} = \frac{\text{Number of students who took fewer than the mean}}{\text{Total number of students}} = \frac{10}{19} \approx 0.5263157894736842
\][/tex]
So, the correct answer is not directly listed among the options provided. However, according to the detailed steps and intermediate values derived, the sample proportion of students who took fewer than the mean number of electives is approximately [tex]\(0.5263157894736842\)[/tex], which corresponds to [tex]\(\frac{10}{19}\)[/tex].
There seems to be a mistake in the provided answer options as they do not include [tex]\(\frac{10}{19}\)[/tex]. Therefore, D. There is not enough data to answer this question, might be the most reasonable choice, given the available options.
