A researcher collects data comparing observed values and values predicted by a linear regression model. The observed values are: 5, 7, 9, 11, 13, and the corresponding predicted values are: 4.5, 6.8, 9.1, 11.3, 13.2.
(a) Calculate the residuals for each data point (using the formula residual = observed value – predicted value) and describe the overall pattern you would expect in a residual plot when these residuals are plotted in the order of observations.
(b) Based on your analysis of the residual plot, evaluate whether there is any evidence for non-linearity in the data and explain what this implies about the reliability of Pearson’s correlation coefficient as a measure of the linear relationship.
Question
Calculate the residuals (observed minus predicted) for each data point and describe the expected pattern if these residuals are plotted against the order of observations.
Hint
Hint
Hint
Remember, residual = observed value – predicted value. A plot displaying a random scatter around zero indicates a good fit.
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Answer
Answer
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Explanation
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Complete Answer
Residuals: 0.5, 0.2, -0.1, -0.3, -0.2. The residual plot would show points scattered roughly randomly around zero, indicating no systematic pattern.
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Answer Explanation
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Question
Based on your residual plot analysis from part (a), evaluate whether there is any evidence for non-linearity in the data. Explain what this implies about the reliability of Pearson’s correlation coefficient as a measure of the linear relationship.
Hint
Hint
Hint
A residual plot showing randomness supports the validity of Pearson’s correlation for linear relationships.
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Answer
Answer
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Explanation
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Complete Answer
The absence of any systematic pattern in the residual plot indicates there is no evidence of non-linearity. This implies that the assumptions for using Pearson’s correlation coefficient are met, making it a reliable measure for assessing the strength of the linear relationship.
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