S3 June 2015 Q1
1. A mobile library has 160 books for children on its records. The librarian believes that books with fewer pages are borrowed more often. He takes a random sample of 10 books for children.
The librarian ranked the 10 books according to how often they had been borrowed, with 1 for the book borrowed the most and 10 for the book borrowed the least. He also recorded the number of pages in each book. The results are in the table below.
| Book | \(A\) | \(B\) | \(C\) | \(D\) | \(E\) | \(F\) | \(G\) | \(H\) | \(I\) | \(J\) |
| Borrowing rank | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Number of pages | 50 | 212 | 115 | 80 | 301 | 90 | 356 | 283 | 152 | 317 |
| Scheme | Marks |
|---|---|
| Label all the books from 1 – 160 (o.e.) | B1 |
| Use random numbers to select the 10 books | B1 |
| (2) |
Notes
1st B1 for labelling/numbering/listing/using sampling frame of all 160 books
2nd B1 for use of random numbers/selection and mentioning the number 10
| Scheme | Marks | ||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| M1 M1 | ||||||||||||||||||||||||||||||||||||||||||||
| \(r_s = 1 - \dfrac{6 \times 66}{10(100 - 1)}\), \([= 1 - 0.4]\) \(= 0.6\) 0.6 | M1,A1 | ||||||||||||||||||||||||||||||||||||||||||||
| (4) |
Notes
1st M1 for an attempt to rank the number of pages (at least 4 correct) Allow reverse ranks
2nd M1 for attempt at \(d^2\) row (may be implied by sight of \(\sum d^2 = 66\) or 264 for reverse ranks)
3rd M1 for use of the correct formula, follow through their \(\sum d^2\) if clearly stated
If answer is not correct, a correct expression is required.
A1 for 0.6 (or \(-0.6\) for reverse ranks)
| Scheme | Marks |
|---|---|
| \(\mathrm{H_0}\): \(\rho = 0 \quad\)\(\mathrm{H_1}\): \(\rho \gt 0\) | B1 |
| Critical value is 0.5636 | B1 |
| 0.6 > cv so significant result and sufficient evidence to reject \(\mathrm{H_0}\) There is support for the librarian’s belief or there is evidence of a correlation between the number of pages in a book and the number of times it is borrowed. | B1ft |
| (3) | |
| (9 marks) |
Notes
1st B1 for both hypotheses in terms of \(\rho\), one tail \(\mathrm{H_1}\) (compatible with ranks) Allow use of \(\rho_s\)
Hypotheses just in words e.g. “no correlation” score B0.
2nd B1 for cv of 0.5636 [If they have a two tail \(\mathrm{H_1}\) then allow 0.6485]
Allow \(\pm\) for reverse ranking but must be same sign as \(r_s\)
If hypotheses are the wrong way around this must be B0 but 3rd B1 is possible.
3rd B1ft for a correct contextualised comment. Must mention “librarian” (or he) or “number of pages” and “borrowing”
Follow through their \(r_s\) and their cv (provided it is \(|\text{cv}| \lt 1\))
Don’t insist on the word “positive” or “negative” for a one-tailed test
Use of “association” is B0
Independent of 1st B1 so if \(|r_s| \gt |\text{cv}|\) must say there is sufficient evidence of …….(o.e.) and if \(|r_s| \lt |\text{cv}|\) must say insufficient evidence of … (o.e.) regardless of their hypotheses