To find the expected value for a student to spend on lunch each day, follow these steps:
1. Understand the Data:
We have a survey that tells us how much money different numbers of students spend on lunch each day.
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Number of Students} & \text{Dollars Spent on Lunch Each Day} \\
\hline
2 & \$10 \\
\hline
1 & \$8 \\
\hline
12 & \$6 \\
\hline
23 & \$5 \\
\hline
8 & \$4 \\
\hline
4 & \$3 \\
\hline
\end{array}
\][/tex]
2. Total Number of Students (N):
Add the number of students surveyed.
[tex]\[
N = 2 + 1 + 12 + 23 + 8 + 4 = 50
\][/tex]
3. Calculate the Weighted Sum of Money Spent:
Multiply the number of students by the dollars they spent and then sum these products.
[tex]\[
\text{Weighted Sum} = (2 \times 10) + (1 \times 8) + (12 \times 6) + (23 \times 5) + (8 \times 4) + (4 \times 3)
\][/tex]
4. Perform the Multiplications:
[tex]\[
\text{Weighted Sum} = 20 + 8 + 72 + 115 + 32 + 12 = 259
\][/tex]
5. Expected Value Calculation:
The expected value [tex]\(E\)[/tex] is calculated by dividing the weighted sum of money spent by the total number of students.
[tex]\[
E = \frac{\text{Weighted Sum}}{N} = \frac{259}{50} = 5.18
\][/tex]
Therefore, the expected value for a student to spend on lunch each day is [tex]\(\$5.18\)[/tex].
So, the correct answer is:
[tex]\[
\$5.18
\][/tex]
Hence, the expected value for a student to spend on lunch each day is [tex]\(\$5.18\)[/tex].
