Question 1180053: Graph the following residuals, and indicate which of the assumptions underlying regression appear to be in jeopardy on the basis of the graph:
x= 213 216 227 229 237 247 263
y- ŷ= - 11 - 5 - 2 - 1 6 10 12
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to graph the residuals and analyze the assumptions:
1. **Create a Scatterplot:**
* Plot the x-values on the horizontal axis and the residuals (y - ŷ) on the vertical axis.
2. **Analyze the Scatterplot:**
* **Linearity:** Look for a random scatter of points around the horizontal axis (y = 0). If there is a clear pattern (e.g., a curve), the linearity assumption might be violated.
* **Constant Variance (Homoscedasticity):** The spread of the residuals should be roughly constant across all x-values. If the spread increases or decreases as x increases, the constant variance assumption might be violated.
* **Normality:** While harder to assess visually, look for a roughly symmetric distribution of residuals around the horizontal axis. If the distribution is heavily skewed or has outliers, the normality assumption might be in jeopardy.
* **Independence:** This assumption is difficult to assess from a residual plot alone. However, if there is a clear pattern (e.g., a cyclical trend), it could suggest that the residuals are not independent.
**Analysis of Your Data:**
Based on the given data, here's what we can observe from a scatterplot of the residuals:
* **Linearity:** There seems to be a slight upward trend in the residuals as x increases. This could indicate a mild violation of the linearity assumption.
* **Constant Variance:** The spread of residuals appears to be relatively constant across the x-values, suggesting that the constant variance assumption is likely met.
* **Normality:** With only seven data points, it's difficult to assess normality visually. However, there are no extreme outliers or strong skewness.
* **Independence:** There is no clear pattern suggesting a violation of independence.
**Conclusion:**
Based on the residual plot, the linearity assumption might be slightly in jeopardy. However, with such a small sample size, it's difficult to draw definitive conclusions. Further analysis with more data points might be needed to confirm any violations.
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