S4 June 2012 Q4
4. A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku.
5.0 4.5 4.7 5.3 5.2 4.1 5.3 4.8 5.5 4.6
Given that the times to complete the Sudoku follow a normal distribution,
of the times taken by people to complete the Sudoku. (13)
The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
| Scheme | Marks |
|---|---|
| \(\bar{x} = 4.9\) | B1 |
| \(s = \sqrt{0.191..}\) (0.437...) (NB: \(\Sigma x = 49;\ \Sigma x^2 = 241.82\)) | B1 |
| (i) 95% confidence interval is given by \(4.9 \pm 2.262 \times \sqrt{\dfrac{0.191..}{10}}\) | M1A1ft B1 |
| i.e: \((4.587\ldots,\ 5.212\ \ldots)\) | A1 A1 |
| (ii) 95% confidence interval is given by \(\dfrac{9 \times 0.437\ldots^2}{19.023} \lt \sigma^2 \lt \dfrac{9 \times 0.437\ldots^2}{2.7}\) use of \(\dfrac{(n-1)s^2}{\chi^2_{n-1}}\) | M1B1B1A1 |
| i.e; \((0.0904,\ 0.63704)\) | A1 A1 |
| (13) |
Notes
B1 B1 may be implied by correct a correct answer to (i) or (ii)
(i) M1 - “their 4.9” \(\pm\) t value \(\times \sqrt{\dfrac{\text{their }0.191..}{10}}\)
A1ft - “their 4.9” \(\pm 2.262 \times \sqrt{\dfrac{\text{their }0.191..}{10}}\)
B1 2.262
A1 either correct to 3sf or better or both correct to 2sf or better
A1 both correct to 3sf or better
(ii) M1 – writing and attempting to use \(\dfrac{(n-1)s^2}{\chi^2_{n-1}}\) or may be implied by correct formula used with their 0.437
B1 19.023
B1 2.7
A1ft follow through their 0.437 and two chi squared values
A1 either correct to 2sf or better
A1 awrt (0.09, 0.637)
| Scheme | Marks |
|---|---|
| 5 lies inside the confidence interval | B1ft |
| \(0.49(0.7^2)\) lies inside the confidence interval | B1ft |
| Yes it does meet the time requirement | B1 ft |
| (3) | |
| (16 marks) |
Notes
For the second B1. If both 0.7 and 0.49 lie in interval they must state variance = 0.49 or the interval for standard deviation.
For the third B1 their must not be two conflicting conclusions unless they give just one overall as well.