7 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.
| Grade | A* | A | B | C | D | E | F | G | U |
| Points | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1 | 0 |
- Calculate the mean GCSE score, \(X\), of a student who has the following GCSE grades:
$$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$
60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.
| Mean GCSE score | Number of students |
| \(4.5 \leqslant X < 5.5\) | 8 |
| \(5.5 \leqslant X < 6.0\) | 14 |
| \(6.0 \leqslant X < 6.5\) | 19 |
| \(6.5 \leqslant X < 7.0\) | 13 |
| \(7.0 \leqslant X \leqslant 8.0\) | 6 |
- Draw a histogram to illustrate this information.
- Calculate estimates of the sample mean and the sample standard deviation.
The scoring system for AS grades is shown in the table below.
| AS Grade | A | B | C | D | E | U |
| Score | 60 | 50 | 40 | 30 | 20 | 0 |
- What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
- What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
- Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.