Use the image and table to answer the question.

Question

An illustration shows a scatterplot titled Number of Laps the Swim Team Completes Per Week. The x-axis shows weeks ranging from 1 through 10 in increments of one. The y-axis shows number of laps and ranges from 100 to 1,000 in 100 unit increments.

Week Number of Laps
1 50
2 150
3 200
4 300
5 500
6 600
7 700
8 800
Which of the following is an appropriate regression model, rounded to the nearest tenth, given the bivariate dataset?

(1 point)
Responses

f(x)∼111.9x−91.1
f left parenthesis x right parenthesis is similar to 111.9 x minus 91.1

f(x)∼111.9x
f left parenthesis x right parenthesis is similar to 111.9 x

f(x)∼−111.9x−91.1
f left parenthesis x right parenthesis is similar to negative 111.9 x minus 91.1

f(x)∼111.9x+50
f left parenthesis x right parenthesis is similar to 111.9 x plus 50

Answer 1

To determine the appropriate regression model based on the provided data, we can observe the trend in the number of laps over the weeks.

From the table of data:

• Week 1: 50 laps
• Week 2: 150 laps
• Week 3: 200 laps
• Week 4: 300 laps
• Week 5: 500 laps
• Week 6: 600 laps
• Week 7: 700 laps
• Week 8: 800 laps

By analyzing the differences between consecutive weeks, we see that the number of laps increases significantly. Given that the number of laps appears to be increasing roughly 100 laps or more for some intervals, we are likely looking at a linear regression model.

If we examine the options given:

1. \( f(x) \sim 111.9x - 91.1 \)
2. \( f(x) \sim 111.9x \)
3. \( f(x) \sim -111.9x - 91.1 \)
4. \( f(x) \sim 111.9x + 50 \)

The first option suggests a line with a y-intercept, while the second one suggests a direct proportional relationship. The third option indicates a negative slope, which does not make sense given that the number of laps increases over time, and the fourth one suggests a positive intercept yet is also linear.

Given the increasing nature of the number of laps, the approaches are likely looking at some positive slope. The simplest linear model is \( f(x) \sim 111.9x \), which would suggest that for each week, there is a strong positive relationship of roughly 111.9 laps per week.

Thus, the appropriate regression model rounded to the nearest tenth is:

f(x) ∼ 111.9x.