Question
Two soccer teams are tied for the league championship going into the final game of the season. The lists below show the number of goals scored by each team in its last 8 games.
Team A: 1, 2, 3, 3, 3, 3, 4, 5
Team B: 2, 2, 2, 3, 3, 4, 4, 4
Which of the following statements best describes graphs of the teams’ goals scored data?
(1 point)
Responses
There will be some overlap because the difference of the means is small and the quotient of the mean absolute deviations is close to 1.
There will be some overlap because the difference of the means is small and the quotient of the mean absolute deviations is close to 1.
There will be some overlap because the difference of the means is 0, but the quotient of the mean absolute deviations is not 1.
There will be some overlap because the difference of the means is 0, but the quotient of the mean absolute deviations is not 1.
There will be almost complete overlap because the difference of the means is 0 and the quotient of the mean absolute deviations is 1.
There will be almost complete overlap because the difference of the means is 0 and the quotient of the mean absolute deviations is 1.
There will be no overlap because the difference of the means is much greater than the quotient of the mean absolute deviations.
Team A: 1, 2, 3, 3, 3, 3, 4, 5
Team B: 2, 2, 2, 3, 3, 4, 4, 4
Which of the following statements best describes graphs of the teams’ goals scored data?
(1 point)
Responses
There will be some overlap because the difference of the means is small and the quotient of the mean absolute deviations is close to 1.
There will be some overlap because the difference of the means is small and the quotient of the mean absolute deviations is close to 1.
There will be some overlap because the difference of the means is 0, but the quotient of the mean absolute deviations is not 1.
There will be some overlap because the difference of the means is 0, but the quotient of the mean absolute deviations is not 1.
There will be almost complete overlap because the difference of the means is 0 and the quotient of the mean absolute deviations is 1.
There will be almost complete overlap because the difference of the means is 0 and the quotient of the mean absolute deviations is 1.
There will be no overlap because the difference of the means is much greater than the quotient of the mean absolute deviations.
Answers
GPT-4o mini
To analyze the goals scored by Team A and Team B, we can calculate their means and mean absolute deviations.
**Team A's Goals: 1, 2, 3, 3, 3, 3, 4, 5**
1. **Mean of Team A:**
\[
\text{Mean} = \frac{1 + 2 + 3 + 3 + 3 + 3 + 4 + 5}{8} = \frac{24}{8} = 3
\]
2. **Mean Absolute Deviation of Team A:**
\[
\text{Absolute Deviations} = |1-3|, |2-3|, |3-3|, |3-3|, |3-3|, |3-3|, |4-3|, |5-3| = 2, 1, 0, 0, 0, 0, 1, 2
\]
\[
\text{Mean Absolute Deviation} = \frac{2 + 1 + 0 + 0 + 0 + 0 + 1 + 2}{8} = \frac{6}{8} = 0.75
\]
**Team B's Goals: 2, 2, 2, 3, 3, 4, 4, 4**
1. **Mean of Team B:**
\[
\text{Mean} = \frac{2 + 2 + 2 + 3 + 3 + 4 + 4 + 4}{8} = \frac{22}{8} = 2.75
\]
2. **Mean Absolute Deviation of Team B:**
\[
\text{Absolute Deviations} = |2-2.75|, |2-2.75|, |2-2.75|, |3-2.75|, |3-2.75|, |4-2.75|, |4-2.75|, |4-2.75| = 0.75, 0.75, 0.75, 0.25, 0.25, 1.25, 1.25, 1.25
\]
\[
\text{Mean Absolute Deviation} = \frac{0.75 + 0.75 + 0.75 + 0.25 + 0.25 + 1.25 + 1.25 + 1.25}{8} = \frac{6.5}{8} = 0.8125
\]
**Summary of Results:**
- **Mean of Team A:** 3
- **Mean of Team B:** 2.75
- **Difference of the Means:** 3 - 2.75 = 0.25
- **Mean Absolute Deviation of Team A:** 0.75
- **Mean Absolute Deviation of Team B:** 0.8125
- **Quotient of the Mean Absolute Deviations:**
\[
\frac{0.75}{0.8125} \approx 0.9231
\]
Based on this analysis, we find that:
- The difference of the means (0.25) is small.
- The quotient of the mean absolute deviations is not 1 (it is approximately 0.9231).
Now, looking at the response options:
- The best statement that describes the graphs of the teams’ goals scored data is:
**"There will be some overlap because the difference of the means is small and the quotient of the mean absolute deviations is close to 1."**
**Team A's Goals: 1, 2, 3, 3, 3, 3, 4, 5**
1. **Mean of Team A:**
\[
\text{Mean} = \frac{1 + 2 + 3 + 3 + 3 + 3 + 4 + 5}{8} = \frac{24}{8} = 3
\]
2. **Mean Absolute Deviation of Team A:**
\[
\text{Absolute Deviations} = |1-3|, |2-3|, |3-3|, |3-3|, |3-3|, |3-3|, |4-3|, |5-3| = 2, 1, 0, 0, 0, 0, 1, 2
\]
\[
\text{Mean Absolute Deviation} = \frac{2 + 1 + 0 + 0 + 0 + 0 + 1 + 2}{8} = \frac{6}{8} = 0.75
\]
**Team B's Goals: 2, 2, 2, 3, 3, 4, 4, 4**
1. **Mean of Team B:**
\[
\text{Mean} = \frac{2 + 2 + 2 + 3 + 3 + 4 + 4 + 4}{8} = \frac{22}{8} = 2.75
\]
2. **Mean Absolute Deviation of Team B:**
\[
\text{Absolute Deviations} = |2-2.75|, |2-2.75|, |2-2.75|, |3-2.75|, |3-2.75|, |4-2.75|, |4-2.75|, |4-2.75| = 0.75, 0.75, 0.75, 0.25, 0.25, 1.25, 1.25, 1.25
\]
\[
\text{Mean Absolute Deviation} = \frac{0.75 + 0.75 + 0.75 + 0.25 + 0.25 + 1.25 + 1.25 + 1.25}{8} = \frac{6.5}{8} = 0.8125
\]
**Summary of Results:**
- **Mean of Team A:** 3
- **Mean of Team B:** 2.75
- **Difference of the Means:** 3 - 2.75 = 0.25
- **Mean Absolute Deviation of Team A:** 0.75
- **Mean Absolute Deviation of Team B:** 0.8125
- **Quotient of the Mean Absolute Deviations:**
\[
\frac{0.75}{0.8125} \approx 0.9231
\]
Based on this analysis, we find that:
- The difference of the means (0.25) is small.
- The quotient of the mean absolute deviations is not 1 (it is approximately 0.9231).
Now, looking at the response options:
- The best statement that describes the graphs of the teams’ goals scored data is:
**"There will be some overlap because the difference of the means is small and the quotient of the mean absolute deviations is close to 1."**