Question

question 7
1 pts
there is a linear and positive relationship between the distance of a hike (measured in miles) and the time it takes to complete the hike (in minutes). after observing many hikes, a researcher constructs a regression equation, as shown below, in order to predict time to complete a hike from the distance of the hike.
\\(\hat{y} = -12.4 + 26.1x\\)
which one of the following statements is the most accurate interpretation of this information?
\\(\bigcirc\\) because negative times will be predicted for very short hikes, a mistake was likely made when constructing the regression equation.
\\(\bigcirc\\) the regression line will eventually cross the vertical axis at a negative value.
\\(\bigcirc\\) although possible, its rare for a regression equation like this one to have a positive slope and a negative intercept.
\\(\bigcirc\\) the regression equation will only be valid when used to make predictions for hikes with long distances.
\\(\bigcirc\\) to obtain a negative intercept, the researcher must have accidentally switched the \x\ and \y\ variables.

Explanation:

1. Analyze Option 1: A negative time prediction for very short hikes doesn't mean a mistake. Regression models extrapolate, and short hikes (x near 0) might give negative ŷ, but the model is for the observed range, not necessarily a mistake. So this is incorrect.
2. Analyze Option 2: The regression equation is \(\hat{y} = -12.4 + 26.1x\). The y - intercept (when \(x = 0\)) is \(- 12.4\). So the regression line crosses the vertical (y - axis, when \(x = 0\)) at \(y=-12.4\), a negative value. This is correct.
3. Analyze Option 3: It is possible to have a positive slope and negative intercept. For example, in a situation where the independent variable (distance) has a positive relationship with the dependent variable (time), but when distance is 0, the predicted time is negative (which may not be meaningful in the real - world context of time but is a mathematical result of the regression). So saying it's rare is incorrect.
4. Analyze Option 4: The regression equation's validity is not only for long - distance hikes. It can be valid for the range of distances on which it was built, not just long distances. So this is incorrect.
5. Analyze Option 5: Switching x and y variables would change the interpretation, but a negative intercept can occur without switching variables. For example, if we are predicting time from distance, and when distance is 0, the model gives a negative time (which is a mathematical outcome, not an error of variable switching). So this is incorrect.

Answer:

B. The regression line will eventually cross the vertical axis at a negative value.