Would we use the regression line to predict growth in per capita income for cities where 70% of residents have at least 4 years of college?

Question

BlackTom AI · Accepted Answer

The question asks about using the regression line to predict growth in per capita income for cities that meet a specific condition (70% of residents with at least 4 years of college).
Since the answer options are not provided in the prompt, we can still reason about the underlying concept and the given answer label, which is "No".
First, consider what a regression line represents: it summarizes the average relationship between two variables (for example, college attainment and growth in per capita income) across the data used to fit the model. It provides predictions for observations drawn from that same population and under similar conditions.
Next, applying the regression line to predict growth for a subset of cities that meet a particular criterion (here, 70% college attainment) is only valid if the subset was part of the data used to fit the model, or if we explicitly re-fit or adjust the model for that subset. If the original regression was built on all cities without conditioning on the 70% threshold, using it directly to predict growth for this conditioned group may be inappropriate because the relationship could change when focusing on that subset. In other words, the conditional subset could exhibit a different intercept or slope, and the unconditional regression line may misrepresent the true relationship within that subset.
Furthermore, statistically, predicting for a subset requires either:
- using the same model but applying it to observations that came from the same population (which preserves applicability), or
- re-estimating the model using only data from that subset to obtain a conditional regression line that accurately reflects the relationship within cities meeting the 70% threshold.
If none of these steps are taken, the straight regression line from the full data is not appropriately guaranteed to predict growth for cities with that college attainment criterion.
Therefore, stating that we would not use the regression line to make that prediction (without re-estimation or conditioning) aligns with sound statistical reasoning. The argument hinges on the need for the model to reflect the specific subset or to be validated for it, rather than blindly applying the overall model.
Note: If the dataset or instructions intended that a separate option exists asserting "Yes" or "No" with a justification, the key issue would be whether the model accounts for conditioning on the 70% threshold. In the absence of such detail, the conservative conclusion is that you should not rely on the original regression line for that conditioned prediction without additional modeling steps.

STAT 1450.02 SP2025 (28132) Checkpoint 5.6 & 5.7

Would we use the regression line to predict growth in per capita income for cities where 70% of residents have at least 4 years of college?

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STAT 1450.02 SP2025 (28132) Checkpoint 5.6 & 5.7

Would we use the regression line to predict growth in per capita income for cities where 70% of residents have at least 4 years of college?

查看解析

登录即可查看完整答案

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Consider the estimated regression model Y_hat = 0.5 + 1.2X1. This model was estimated on a data set where X1 has a range of values between 8 and 34. Should we use the model to predict Y_hat=2.9 = 0.5 + 1.22 for X1=2?