a. Linear Equation of Best Fit:

The linear equation of best fit for the data is:

$\text{Calories/lap} = 0.595 \cdot \text{Speed (mph)} + 34.571$

This equation suggests that for every 1 mph increase in speed, the calories burned per lap increase by approximately 0.595 calories.

Comparing the model to the data, the model does not fit the data well due to the low correlation coefficient (0.273), indicating that the linear relationship is weak. Additionally, the p-value is high (0.513), implying that the linear model may not be statistically significant.

b. Significance of the y-Intercept:

The y-intercept is 34.571. This represents the estimated calories burned per lap when the speed is 0 mph. In this context, it could mean that at zero speed, there is still a baseline caloric burn, but this might not be practically significant since you wouldn't burn many calories without movement.

c. Slope and its Meaning:

The slope is 0.595, which means that for every 1 mph increase in speed, the calories burned per lap increase by approximately 0.595 calories. It indicates a small positive relationship between speed and calories burned.

d. x-Intercept and its Meaning:

To find the x-intercept (when calories burned is 0), we set the equation equal to zero:

$0 = 0.595 \cdot \text{Speed} + 34.571$

$\text{Speed} = -\frac{34.571}{0.595} \approx -58.07 \text{ mph}$

This x-intercept is negative, which does not make practical sense in this context since speed cannot be negative. Therefore, the x-intercept does not have a meaningful interpretation for this data set.

e. Correlation Coefficient:

The correlation coefficient is approximately 0.273. This indicates a weak positive relationship between speed and calories burned. In other words, as speed increases, calories burned tend to increase, but the relationship is not strong.

Would you like more details on any part of the explanation?

Here are 5 related questions to expand on this:

1. How can we improve the accuracy of this model?
2. What are other statistical methods to assess the fit of a model?
3. Can a polynomial model provide a better fit for this data?
4. How does the p-value affect the reliability of the model?
5. What does the standard error of the slope indicate about the variability?

Tip: A low correlation coefficient suggests that other factors may be influencing the data. You might explore nonlinear models or additional variables.