- Students on a psychology course were given a pre-test at the start of the course and a final exam at the end of the course. The teacher recorded the number of marks achieved on the pre-test, \(p\), and the number of marks achieved on the final exam, \(f\), for 34 students and displayed them on the scatter diagram.
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The equation of the least squares regression line for these data is found to be
$$f = 10.8 + 0.748 p$$
For these students, the mean number of marks on the pre-test is 62.4
- Use the regression model to find the mean number of marks on the final exam.
- Give an interpretation of the gradient of the regression line.
Considering the equation of the regression line, Priya says that she would expect someone who scored 0 marks on the pre-test to score 10.8 marks on the final exam.
- Comment on the reliability of Priya's statement.
- Write down the number of marks achieved on the final exam for the student who exceeded the expectation of the regression model by the largest number of marks.
- Find the range of values of \(p\) for which this regression model, \(f = 10.8 + 0.748 p\), predicts a greater number of marks on the final exam than on the pre-test.
Later the teacher discovers an error in the recorded data. The student who achieved a score of 98 on the pre-test, scored 92 not 29 on the final exam.
The summary statistics used for the model \(f = 10.8 + 0.748 p\) are corrected to include this information and a new least squares regression line is found.
Given the original summary statistics were,
$$n = 34 \quad \sum p = 2120 \quad \sum p f = 133486 \quad \mathrm {~S} _ { p p } = 15573.76 \quad \mathrm {~S} _ { p f } = 11648.35$$
- calculate the gradient of the new regression line. Show your working clearly.