The value of a bicycle was recorded over a period of five years. Based on how the plot of the

Question

residuals might look, would a linear function that passes through (0,250) and (4, 25) be a good fit
for the data shown?

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250

§ 200

150

100

50

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

Yes, the residuals would tend to be one-sided.

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No, the residuals would be randomly distributed.

0

1

2

3

4

5 x
Age of Bicycle (in years)

Answer 1

To determine whether a linear function that passes through the points (0, 250) and (4, 25) would be a good fit for the data, we need to consider the expected residuals and their distribution.

1. Residuals Definition: The residuals are the differences between the observed values and the values predicted by the linear model at each point. If the model is a good fit, the residuals should be randomly distributed around zero.

2. Key Points:

   • The function is decreasing from year 0 (value = 250) to year 4 (value = 25), suggesting that as the age of the bicycle increases, its value decreases.
   • If the actual values of the bicycle for each age do not follow this linear trend (for example, they might decrease more quickly at first and then level off, or they could have a different non-linear pattern), the residuals would likely exhibit a pattern, rather than being randomly scattered.

3. One-sided Residuals: If the model consistently underestimates or overestimates the actual values, the residuals would tend to be on one side (either mostly positive or mostly negative). This would suggest a poor fit.

4. Random Distribution of Residuals: Ideally, a good linear fit would result in residuals that are randomly scattered, both above and below the zero line.

Based on your response options, you should consider:

• If the residuals tend to be one-sided, that suggests the model is not capturing the values correctly. So in that case, the answer would be "No, the residuals would tend to be one-sided."

• If the residuals can be described as scattered around zero (not showing a clear pattern), then you would say "Yes, the residuals would tend to be randomly distributed."

Considering the lack of detailed information about the actual values at each year, you cannot definitively conclude whether the linear model is a good fit just based on the points given.

If the values at years 1, 2, 3, and 5 align well with what the linear trend suggests, the linear function would be appropriate and produce randomly distributed residuals. However, if, for example, the value starts above 250 initially or decreases at an uneven rate, then the residuals would not be randomly scattered.

In conclusion, without more context about the actual values of the bicycle over the years, it’s impossible to definitively choose one option. Generally, the correct answer should be aligned with the actual observed data’s behavior relative to the predicted values. Therefore, if you observe a clear pattern in residuals (one-sided) based on insights from the plot or additional information, choose the appropriate answer accordingly.