Math Problem Statement
Solution
The problem involves analyzing a set of student scores for two tests (Test A and Test B), and working with the regression line , where represents the score on Test A and the score on Test B. Let’s break this down and solve each part:
(a) Find the value of Pearson’s product-moment correlation coefficient .
The correlation coefficient can be derived from the regression line if additional statistical summaries (e.g., variances) are known. Based on the data given, further calculation is needed for .
(b) Why is the regression method inappropriate for Paulo?
The teacher used Paulo's score of on Test A to estimate his score on Test B using the regression equation. However, this method is not appropriate because:
- Extrapolation: Paulo's score of lies far outside the range of -values in the table. Extrapolating the regression line beyond the observed range often leads to inaccurate predictions.
- Regression assumptions: The linear relationship assumed by the regression equation may not hold for values outside the dataset.
(c) Why is the regression method inappropriate for Giovanni?
For Giovanni, the teacher used on Test B to estimate on Test A. This method is inappropriate because:
- Regression direction: The regression equation provided predicts from , not from . Estimating from would require the inverse regression equation, which isn't derived here.
- Error in estimates: The correlation is not perfect (), so estimates are subject to error. Using a regression line for inverse predictions amplifies this error.
(d) Show that Giovanni’s Test A score is 86.
We substitute Giovanni's Test B score into the regression equation:
Rearranging for :
Substitute :
To the nearest integer, .
This matches the teacher’s estimate, but as mentioned earlier, it’s subject to errors for inverse predictions.
Would you like a deeper explanation of the calculations or how regression works?
Relative Questions:
- How is Pearson’s correlation coefficient calculated, and what does it represent?
- What are the common assumptions made in linear regression analysis?
- What happens when you extrapolate a regression line outside its observed range?
- Why is the inverse regression equation different from the original regression line?
- Can we calculate (the coefficient of determination) from this dataset?
Tip:
In regression, avoid using predictions outside the observed data range or the wrong direction of prediction, as it can significantly distort results.
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Pearson's Correlation Coefficient
Extrapolation
Inverse Regression
Formulas
Regression equation: y = 0.822x + 18.4
Pearson's Correlation Coefficient r formula
Inverse regression: x = (y - b) / m
Theorems
Properties of Linear Regression
Assumptions of Correlation
Suitable Grade Level
Grades 10-12
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