Ask a New Question
Menu

the table dispalys the number of laps completed each week over a 8 week period of time. 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−91.1 f left parenthesis x right parenthesis is similar to negative 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+50

1 answer

To determine the appropriate regression model for the given dataset, we first need to analyze the data presented:

  • 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

We can observe the number of laps grows as the weeks progress. A good approach to model this type of data is to perform a linear regression since the relationship seems to exhibit a linear trend.

Calculating the Slope and Intercept

We can first calculate the slope (m) and the intercept (b) of a linear regression line in the form of \( f(x) = mx + b \).

  • The average number of laps per week will be used to find the slope:
    • The total laps after 8 weeks = \( 50 + 150 + 200 + 300 + 500 + 600 + 700 + 800 = 3350 \)
    • The total number of weeks = 8
    • Average laps per week (total laps / number of weeks) = \( 3350 / 8 = 418.75 \).

The increase in laps over the weeks demonstrates a linear relationship. Let’s look for an increase in laps over the weeks:

  • Week 1 to 2: Increased by 100 laps (150 - 50)
  • Week 2 to 3: Increased by 50 laps (200 - 150)
  • Week 3 to 4: Increased by 100 laps (300 - 200)
  • Week 4 to 5: Increased by 200 laps (500 - 300)
  • Week 5 to 6: Increased by 100 laps (600 - 500)
  • Week 6 to 7: Increased by 100 laps (700 - 600)
  • Week 7 to 8: Increased by 100 laps (800 - 700)

The average increase seems to be around 111.9 laps per week across the weeks following an initial period.

Comparing the Options

After calculating the average increase, we can review the given options:

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

The first model includes an intercept, which may be consistent with the starting point, while the rest vary based on whether they include negative signs or other constants.

Since we are looking for a model without a negative slope and the function must be linear with a positive intercept, the closest option is:

Answer:

\( f(x) \sim 111.9x - 91.1 \) is the best fit based on representing the increment of laps over weeks against time.