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Which of the following is an appropriate regression model weeks 1, 2 ,3, 4 number of laps 50,150 ,200, 300

F of x equals 111.9 X f of x = -111.9 X - 91.9 f of x equals 111.9 x - 91.1 f of x = 111.9 x + 50

1 answer

To determine the appropriate regression model for the given data, we need to analyze the relationship between the independent variable (weeks) and the dependent variable (number of laps). The data is as follows:

  • Week 1: 50 laps
  • Week 2: 150 laps
  • Week 3: 200 laps
  • Week 4: 300 laps

Let's denote the number of weeks as \( x \) and the number of laps as \( y \). The pairs are:

  • \( (1, 50) \)
  • \( (2, 150) \)
  • \( (3, 200) \)
  • \( (4, 300) \)

Step 1: Analyze the Data

The increase in the number of laps per week appears to be non-linear initially, suggesting a potential quadratic component. However, we can approximate a linear model based on the observed data trend.

Step 2: Determine the Slope

To find the approximate slope (rate of change), we can calculate differences in the laps:

  • From week 1 to week 2: \( 150 - 50 = 100 \)
  • From week 2 to week 3: \( 200 - 150 = 50 \)
  • From week 3 to week 4: \( 300 - 200 = 100 \)

The change is not consistent but suggests an overall upward trend, particularly a strong increase after weeks 1 and 2.

Step 3: Create the Regression Model

Let’s check if any of the proposed functions can fit:

  1. Option 1: \( f(X) = 111.9X \)
  2. Option 2: \( f(X) = -111.9X - 91.9 \)
  3. Option 3: \( f(X) = 111.9X - 91.1 \)
  4. Option 4: \( f(X) = 111.9X + 50 \)

Since the slope for the data appears to be positive, we can disregard options 2 as it has a negative slope.

Test the Options

  • For \( f(X) = 111.9X - 91.1 \):

    • \( f(1) = 111.9(1) - 91.1 = 20.8 \) (not close)
    • \( f(2) = 111.9(2) - 91.1 = 132.7 \) (not close)
    • \( f(3) = 111.9(3) - 91.1 = 244.6 \) (getting closer)
    • \( f(4) = 111.9(4) - 91.1 = 356.5 \) (too high)

  • For \( f(X) = 111.9X + 50 \):

    • \( f(1) = 111.9(1) + 50 = 161.9 \) (too high)
    • \( f(2) = 111.9(2) + 50 = 273.8 \) (close)
    • \( f(3) = 111.9(3) + 50 = 385.7 \) (too high)
    • \( f(4) = 111.9(4) + 50 = 497.6 \) (way too high)

Conclusion

Based on the calculations, none of the proposed functions perfectly fit the data. However, \( f(X) = 111.9X - 91.1 \) comes closer to normalizing in terms of the correct trend, albeit with issues.

Considering the fundamental structure, an updated or corrected model could be formed based on a more nuanced understanding of the data. For a quick answer with the listed options, the closest reasonable option would be \( f(X) = 111.9X - 91.1 \), given its positive slope and tendency to cluster around the upward trend in certain values. However, keep in mind that the selections provided may not fully derive an accurate relation with the data.