Worked solution
Predict a French mark from an English mark
Ten randomly selected students have these French test marks and English test marks .
| (English) | (French) |
|---|---|
| 19 | 20 |
| 42 | 43 |
| 44 | 33 |
| 52 | 56 |
| 51 | 50 |
| 53 | 67 |
| 66 | 73 |
| 56 | 68 |
| 60 | 77 |
| 37 | 43 |
(i) Obtain the product moment correlation between and and comment on it.
(ii) Obtain the regression line of on , correct to 3 decimal places.
(iii) Given that the regression coefficient for on is 0.66, show by calculation that this line is .
(iv) Tom missed the French test but scored 65 in English. Predict a suitable French mark and comment on its reliability.
CJC · Lecture · Example 8
(i) Interpret correlation in context
Answer
(3 s.f.): strong positive linear association.
The source prints , close to 1. In these paired test scores, students with higher English marks tend to have higher French marks. This is an association in the sample, not proof that taking one test causes the other score.
(ii) Round coefficients to three decimal places
Answer
.
The printed full coefficients give . Rounding to the requested 3 decimal places, rather than 3 significant figures, gives the displayed equation. Retain the full values for the later prediction.
(iii) Use the given reverse coefficient and means
Answer
.
The mean point is and lies on both fitted lines. Using the stipulated reverse slope , write , so . The source starts this derivation but leaves the completion blank. The calculation uses the coefficient as given; it does not claim 0.66 is the unrounded fitted slope.
(iv) Predict French from English and assess the range
Answer
Predicted French mark (3 s.f.).
Use on , since French is being predicted from English:
The source calls this reliable because is inside the observed English range (interpolation) and indicates strong positive linear association. These support using the fitted model, but do not guarantee an accurate individual mark. The context should remain comparable to the sampled students; random selection alone does not remove residual variation.
Solution notes
All ten pairs and four source asks retained. Printed correlation, fit, mean point and prediction/reliability rationale preserved. Blank reverse-intercept calculation authored; reliability qualified rather than guaranteed. No printed marks.