Question

the value of a bicycle was recorded over a period of five years. based on how the plot of the residuals might look, would a linear function that passes through (0,250) and (4,25) be a good fit for the data shown? chart: bicycle prices, x-axis labeled age of bicycle (in years), y-axis labeled value of bicycle (in dollars), with data points at (0,250), (1, ~125), (2, ~75), (3, ~30), (4, ~25) options: 1. yes, the residuals would be randomly distributed. 2. no, the residuals would tend to be one-sided. 3. no, the residuals would be randomly distributed. 4. yes, the residuals would tend to be one-sided.

Explanation:

1. First, recall the concept of residuals in linear regression: Residuals are the differences between the observed values and the predicted values from the linear model. A good - fitting linear model should have residuals that are randomly distributed around zero (no pattern, some positive and some negative, and random in terms of magnitude and sign).
2. Analyze the given linear function: The linear function passes through \((0,250)\) and \((4,25)\). The slope \(m=\frac{25 - 250}{4-0}=\frac{- 225}{4}=- 56.25\). The equation of the line is \(y=-56.25x + 250\).
3. Now, look at the scatter - plot of the bicycle prices. The observed data points (the dots in the plot) are all below the line we just found (except maybe the first point at \(x = 0\)). This means that the predicted values from the linear model are greater than the observed values for most of the data points. So the residuals (observed - predicted) will be negative for most of the points (since observed < predicted). In other words, the residuals will tend to be one - sided (mostly negative in this case). A good - fitting linear model should have residuals that are randomly distributed (both positive and negative, with no consistent bias). Since the residuals here are one - sided, the linear function is not a good fit.

Answer:

B. No, the residuals would tend to be one - sided.