Question
Sam surveyed a sample of 10 seventh graders, asking how much money they saved and spent on average each week. He recorded his results in these line plots.
Saved
0:1
5:2
10:5
15:0
20:2
25:0
30:0
35:0
40:0
45:0
50:0
Money (dollars)
Spent
0:0
5:0
10:3
15:2
20:2
25:2
30:0
35:0
40:1
45:0
50:0
Money (dollars)
Complete the table. Round your answers to the nearest cent.
Mean Mean absolute deviation
Saved $10 $
Spent $
$7
Answers
GPT-4o mini
To complete the table, we need to calculate the mean and mean absolute deviation for the "Saved" and "Spent" data.
### For Saved:
1. **Mean:**
- Multiply each amount saved by the frequency, and sum those products:
- \(0 \times 1 = 0\)
- \(5 \times 2 = 10\)
- \(10 \times 5 = 50\)
- \(15 \times 0 = 0\)
- \(20 \times 2 = 40\)
- \(25 \times 0 = 0\)
- \(30 \times 0 = 0\)
- \(35 \times 0 = 0\)
- \(40 \times 0 = 0\)
- \(45 \times 0 = 0\)
- \(50 \times 0 = 0\)
- Total = \(0 + 10 + 50 + 0 + 40 + 0 + 0 + 0 + 0 + 0 + 0 = 100\)
- Total students = \(1 + 2 + 5 + 0 + 2 + 0 + 0 + 0 + 0 + 0 + 0 = 10\)
- Mean = \(\frac{Total}{Total\ students} = \frac{100}{10} = 10\) (as given)
2. **Mean Absolute Deviation:**
- Calculate the absolute deviations from the mean (10):
- \(|0 - 10| = 10\), frequency: 1 → contribution: \(10 \times 1 = 10\)
- \(|5 - 10| = 5\), frequency: 2 → contribution: \(5 \times 2 = 10\)
- \(|10 - 10| = 0\), frequency: 5 → contribution: \(0 \times 5 = 0\)
- \(|15 - 10| = 5\), frequency: 0 → contribution: \(5 \times 0 = 0\)
- \(|20 - 10| = 10\), frequency: 2 → contribution: \(10 \times 2 = 20\)
- All others are \(0\).
- Total absolute deviations = \(10 + 10 + 0 + 0 + 20 = 40\)
- Mean Absolute Deviation = \(\frac{Total\ absolute\ deviations}{Total\ students} = \frac{40}{10} = 4\)
### For Spent:
1. **Mean:**
- Multiply each amount spent by the frequency, and sum those products:
- \(0 \times 0 = 0\)
- \(5 \times 0 = 0\)
- \(10 \times 3 = 30\)
- \(15 \times 2 = 30\)
- \(20 \times 2 = 40\)
- \(25 \times 2 = 50\)
- \(30 \times 0 = 0\)
- \(35 \times 0 = 0\)
- \(40 \times 1 = 40\)
- \(45 \times 0 = 0\)
- \(50 \times 0 = 0\)
- Total = \(0 + 0 + 30 + 30 + 40 + 50 + 0 + 0 + 40 + 0 + 0 = 190\)
- Total students = \(0 + 0 + 3 + 2 + 2 + 2 + 0 + 0 + 1 + 0 + 0 = 10\)
- Mean = \(\frac{Total}{Total\ students} = \frac{190}{10} = 19\)
2. **Mean Absolute Deviation:**
- Calculate the absolute deviations from the mean (19):
- \(|0 - 19| = 19\), frequency: 0 → contribution: \(19 \times 0 = 0\)
- \(|5 - 19| = 14\), frequency: 0 → contribution: \(14 \times 0 = 0\)
- \(|10 - 19| = 9\), frequency: 3 → contribution: \(9 \times 3 = 27\)
- \(|15 - 19| = 4\), frequency: 2 → contribution: \(4 \times 2 = 8\)
- \(|20 - 19| = 1\), frequency: 2 → contribution: \(1 \times 2 = 2\)
- \(|25 - 19| = 6\), frequency: 2 → contribution: \(6 \times 2 = 12\)
- All others are \(0\).
- Total absolute deviations = \(0 + 0 + 27 + 8 + 2 + 12 + 0 + 0 + 0 + 0 + 0 = 49\)
- Mean Absolute Deviation = \(\frac{Total\ absolute\ deviations}{Total\ students} = \frac{49}{10} = 4.9\)
### Completed Table:
| Mean | Mean Absolute Deviation |
|---------|--------------------------|
| Saved | $10.00 | $4.00 |
| Spent | $19.00 | $4.90 |
So the final completed entries to fill in are mean saved \(= 10\), mean absolute deviation saved \(= 4\), mean spent \(= 19\), and mean absolute deviation spent \(= 4.9\).
### For Saved:
1. **Mean:**
- Multiply each amount saved by the frequency, and sum those products:
- \(0 \times 1 = 0\)
- \(5 \times 2 = 10\)
- \(10 \times 5 = 50\)
- \(15 \times 0 = 0\)
- \(20 \times 2 = 40\)
- \(25 \times 0 = 0\)
- \(30 \times 0 = 0\)
- \(35 \times 0 = 0\)
- \(40 \times 0 = 0\)
- \(45 \times 0 = 0\)
- \(50 \times 0 = 0\)
- Total = \(0 + 10 + 50 + 0 + 40 + 0 + 0 + 0 + 0 + 0 + 0 = 100\)
- Total students = \(1 + 2 + 5 + 0 + 2 + 0 + 0 + 0 + 0 + 0 + 0 = 10\)
- Mean = \(\frac{Total}{Total\ students} = \frac{100}{10} = 10\) (as given)
2. **Mean Absolute Deviation:**
- Calculate the absolute deviations from the mean (10):
- \(|0 - 10| = 10\), frequency: 1 → contribution: \(10 \times 1 = 10\)
- \(|5 - 10| = 5\), frequency: 2 → contribution: \(5 \times 2 = 10\)
- \(|10 - 10| = 0\), frequency: 5 → contribution: \(0 \times 5 = 0\)
- \(|15 - 10| = 5\), frequency: 0 → contribution: \(5 \times 0 = 0\)
- \(|20 - 10| = 10\), frequency: 2 → contribution: \(10 \times 2 = 20\)
- All others are \(0\).
- Total absolute deviations = \(10 + 10 + 0 + 0 + 20 = 40\)
- Mean Absolute Deviation = \(\frac{Total\ absolute\ deviations}{Total\ students} = \frac{40}{10} = 4\)
### For Spent:
1. **Mean:**
- Multiply each amount spent by the frequency, and sum those products:
- \(0 \times 0 = 0\)
- \(5 \times 0 = 0\)
- \(10 \times 3 = 30\)
- \(15 \times 2 = 30\)
- \(20 \times 2 = 40\)
- \(25 \times 2 = 50\)
- \(30 \times 0 = 0\)
- \(35 \times 0 = 0\)
- \(40 \times 1 = 40\)
- \(45 \times 0 = 0\)
- \(50 \times 0 = 0\)
- Total = \(0 + 0 + 30 + 30 + 40 + 50 + 0 + 0 + 40 + 0 + 0 = 190\)
- Total students = \(0 + 0 + 3 + 2 + 2 + 2 + 0 + 0 + 1 + 0 + 0 = 10\)
- Mean = \(\frac{Total}{Total\ students} = \frac{190}{10} = 19\)
2. **Mean Absolute Deviation:**
- Calculate the absolute deviations from the mean (19):
- \(|0 - 19| = 19\), frequency: 0 → contribution: \(19 \times 0 = 0\)
- \(|5 - 19| = 14\), frequency: 0 → contribution: \(14 \times 0 = 0\)
- \(|10 - 19| = 9\), frequency: 3 → contribution: \(9 \times 3 = 27\)
- \(|15 - 19| = 4\), frequency: 2 → contribution: \(4 \times 2 = 8\)
- \(|20 - 19| = 1\), frequency: 2 → contribution: \(1 \times 2 = 2\)
- \(|25 - 19| = 6\), frequency: 2 → contribution: \(6 \times 2 = 12\)
- All others are \(0\).
- Total absolute deviations = \(0 + 0 + 27 + 8 + 2 + 12 + 0 + 0 + 0 + 0 + 0 = 49\)
- Mean Absolute Deviation = \(\frac{Total\ absolute\ deviations}{Total\ students} = \frac{49}{10} = 4.9\)
### Completed Table:
| Mean | Mean Absolute Deviation |
|---------|--------------------------|
| Saved | $10.00 | $4.00 |
| Spent | $19.00 | $4.90 |
So the final completed entries to fill in are mean saved \(= 10\), mean absolute deviation saved \(= 4\), mean spent \(= 19\), and mean absolute deviation spent \(= 4.9\).