Question 22·Easy·Two-Variable Data: Models and Scatterplots
A teacher recorded how many hours each of five students studied for a quiz and each student’s quiz score.
| Hours studied, | Quiz score, |
|---|---|
| 1 | 65 |
| 2 | 70 |
| 3 | 75 |
| 4 | 80 |
| 5 | 85 |
Based on the data, what is the best estimate of the average increase in quiz score (in points) for each additional hour studied?
For questions asking for the “average increase” of one variable with respect to another, think of slope: compute using two convenient points, usually the first and last in the table. If the pattern looks linear with equal differences, you can simply take the consistent change in for a 1-unit change in , which is much faster than doing multiple calculations.
Hints
Focus on consecutive rows in the table
Look at what happens to the quiz score when the hours studied go up by 1 each time (from 1 to 2 hours, 2 to 3 hours, etc.).
Compute the change in quiz score
For each 1-hour increase, subtract the earlier score from the later score (for example, ). Are these changes the same each time?
Use rate of change
Once you see the change in score for a 1-hour increase, that change in score is the average increase per hour. You can also confirm it by using the first and last points and computing .
Check against the answer choices
Compare the value you found for the change in score per hour with the choices. Pick the one that matches your computed rate of change.
Desmos Guide
Enter the data as a table
Add a table in Desmos. In the column, enter . In the column, enter so the five data points match the table.
Fit a line to the data
In the expression line, type y1 ~ m x1 + b. Desmos will perform a linear regression and display values for and .
Identify the average increase per hour
Look at the value of from the regression. This is the slope of the best-fit line, which represents the average increase in quiz score (in points) for each additional hour studied. Use that value to choose your answer.
Step-by-step Explanation
Interpret what the question is asking
The question asks for the average increase in quiz score for each additional hour studied. In math terms, this is the rate of change or the slope:
Look at how the score changes as hours increase by 1
Check the change in quiz score between each pair of consecutive rows:
- From to : score goes from to .
- From to : score goes from to .
- From to : score goes from to .
- From to : score goes from to .
In each case, hours increase by , and the score increases by the same number of points each time. This means the rate of change is constant.
Compute the rate of change using two points
Because the pattern is linear (the increase is the same each time), you can use any two points to find the average increase per hour. Using the first and last data points and :
State the final answer and match it to the choices
The fraction simplifies to , so the quiz score increases on average by 5 points for each additional hour studied.
Therefore, the correct choice is B) 5.