To determine the expected value for the amount of money a student spends on lunch each day, we need to calculate the total amount of money spent by all surveyed students and divide it by the total number of students surveyed. Here is a detailed step-by-step solution:
1. Data Collection:
- Number of students and their respective money spent:
- 2 students spend [tex]$10 each
- 1 student spends $[/tex]8
- 12 students spend [tex]$6 each
- 23 students spend $[/tex]5 each
- 8 students spend [tex]$4 each
- 4 students spend $[/tex]3 each
2. Calculate Total Number of Students:
- The total number of students surveyed is found by summing the number of students in each category:
[tex]\[
2 + 1 + 12 + 23 + 8 + 4 = 50 \text{ students}
\][/tex]
3. Calculate Total Money Spent:
- We multiply the number of students by the money they spend to get the total amount of money spent for each group:
- For the 2 students spending [tex]$10:
\[
2 \times 10 = 20
\]
- For the 1 student spending $[/tex]8:
[tex]\[
1 \times 8 = 8
\][/tex]
- For the 12 students spending [tex]$6:
\[
12 \times 6 = 72
\]
- For the 23 students spending $[/tex]5:
[tex]\[
23 \times 5 = 115
\][/tex]
- For the 8 students spending [tex]$4:
\[
8 \times 4 = 32
\]
- For the 4 students spending $[/tex]3:
[tex]\[
4 \times 3 = 12
\][/tex]
- Adding these amounts together gives the total money spent by all students:
[tex]\[
20 + 8 + 72 + 115 + 32 + 12 = 259
\][/tex]
4. Calculate the Expected Value:
- The expected value (average amount spent per student) is the total money spent divided by the total number of students:
[tex]\[
\text{Expected Value} = \frac{259}{50} \approx 5.18
\][/tex]
Therefore, the expected value for a student to spend on lunch each day is [tex]\( \$5.18 \)[/tex].
