Question
Mr. Moore is trying to convince his high school students that it is better to slow down when they are learning to type. He believes that students who type too quickly make too many errors. To prove this point, Mr. Moore has the students take a typing test. He records their average typing speed (in words per minute), x, and the number of errors they made, y.
The line of best fit/regression line of this data is:
y=3.473x−155.317
(5 points)
Using the regression line, about how many errors would be made if the students typed zero words per minute?
For an increase of one word per minute typing speed, the regression line predicts an increase of how many errors?
If a student types at 50 words per minute, how many errors does the regression line predict will be made?
If a student types 70 words per minute, how many errors does the regression line predict will be made?
Does this support Mr. Moore's claim that the slower a student types, the fewer errors will be made?
The line of best fit/regression line of this data is:
y=3.473x−155.317
(5 points)
Using the regression line, about how many errors would be made if the students typed zero words per minute?
For an increase of one word per minute typing speed, the regression line predicts an increase of how many errors?
If a student types at 50 words per minute, how many errors does the regression line predict will be made?
If a student types 70 words per minute, how many errors does the regression line predict will be made?
Does this support Mr. Moore's claim that the slower a student types, the fewer errors will be made?
Answers
GPT-4o mini
Let’s analyze the regression line provided and answer the questions step by step.
1. **Using the regression line, about how many errors would be made if the students typed zero words per minute?**
The regression line is given by the equation:
\[
y = 3.473x - 155.317
\]
If \( x = 0 \) (typing speed of zero words per minute), we substitute \( x \) into the equation:
\[
y = 3.473(0) - 155.317 = -155.317
\]
This means that the predicted number of errors made at a typing speed of zero words per minute would be approximately **-155.32**. However, since the number of errors cannot be negative, this suggests that the model may not be applicable or meaningful at very low speeds.
2. **For an increase of one word per minute typing speed, the regression line predicts an increase of how many errors?**
The slope of the regression line (the coefficient of \( x \)) is 3.473. This indicates that for each increase of one word per minute in typing speed, the model predicts an increase of approximately **3.473 errors**.
3. **If a student types at 50 words per minute, how many errors does the regression line predict will be made?**
Substitute \( x = 50 \) into the regression equation:
\[
y = 3.473(50) - 155.317
\]
First, calculate \( 3.473 \times 50 \):
\[
3.473 \times 50 = 173.65
\]
Now substitute this value back into the equation:
\[
y = 173.65 - 155.317 = 18.333
\]
Therefore, at 50 words per minute, the regression line predicts that approximately **18.33 errors** will be made.
4. **If a student types 70 words per minute, how many errors does the regression line predict will be made?**
Substitute \( x = 70 \) into the regression equation:
\[
y = 3.473(70) - 155.317
\]
First, calculate \( 3.473 \times 70 \):
\[
3.473 \times 70 = 243.11
\]
Now substitute this value back into the equation:
\[
y = 243.11 - 155.317 = 87.793
\]
Therefore, at 70 words per minute, the regression line predicts that approximately **87.79 errors** will be made.
5. **Does this support Mr. Moore's claim that the slower a student types, the fewer errors will be made?**
Mr. Moore's claim suggests that a slower typing speed should result in fewer errors. However, based on the regression analysis, we can see that as the typing speed \( x \) increases (from 50 to 70 words per minute), the number of errors \( y \) also increases (from about 18.33 errors to 87.79 errors).
This trend indicates that, according to the regression line, faster typing correlates with a greater number of errors, contradicting Mr. Moore's claim. Therefore, **the data does not support Mr. Moore's claim** that slower typing leads to fewer errors; rather, it suggests the opposite.
1. **Using the regression line, about how many errors would be made if the students typed zero words per minute?**
The regression line is given by the equation:
\[
y = 3.473x - 155.317
\]
If \( x = 0 \) (typing speed of zero words per minute), we substitute \( x \) into the equation:
\[
y = 3.473(0) - 155.317 = -155.317
\]
This means that the predicted number of errors made at a typing speed of zero words per minute would be approximately **-155.32**. However, since the number of errors cannot be negative, this suggests that the model may not be applicable or meaningful at very low speeds.
2. **For an increase of one word per minute typing speed, the regression line predicts an increase of how many errors?**
The slope of the regression line (the coefficient of \( x \)) is 3.473. This indicates that for each increase of one word per minute in typing speed, the model predicts an increase of approximately **3.473 errors**.
3. **If a student types at 50 words per minute, how many errors does the regression line predict will be made?**
Substitute \( x = 50 \) into the regression equation:
\[
y = 3.473(50) - 155.317
\]
First, calculate \( 3.473 \times 50 \):
\[
3.473 \times 50 = 173.65
\]
Now substitute this value back into the equation:
\[
y = 173.65 - 155.317 = 18.333
\]
Therefore, at 50 words per minute, the regression line predicts that approximately **18.33 errors** will be made.
4. **If a student types 70 words per minute, how many errors does the regression line predict will be made?**
Substitute \( x = 70 \) into the regression equation:
\[
y = 3.473(70) - 155.317
\]
First, calculate \( 3.473 \times 70 \):
\[
3.473 \times 70 = 243.11
\]
Now substitute this value back into the equation:
\[
y = 243.11 - 155.317 = 87.793
\]
Therefore, at 70 words per minute, the regression line predicts that approximately **87.79 errors** will be made.
5. **Does this support Mr. Moore's claim that the slower a student types, the fewer errors will be made?**
Mr. Moore's claim suggests that a slower typing speed should result in fewer errors. However, based on the regression analysis, we can see that as the typing speed \( x \) increases (from 50 to 70 words per minute), the number of errors \( y \) also increases (from about 18.33 errors to 87.79 errors).
This trend indicates that, according to the regression line, faster typing correlates with a greater number of errors, contradicting Mr. Moore's claim. Therefore, **the data does not support Mr. Moore's claim** that slower typing leads to fewer errors; rather, it suggests the opposite.