S4 June 2016 Q7
7. The times taken to travel to school by sixth form students are normally distributed. A head teacher records the times taken to travel to school, in minutes, of a random sample of 10 sixth form students from her school.
Based on this sample, the 95% confidence interval for the mean time taken to travel to school for sixth form students from her school is
\[[28.5,\ 48.7]\]Calculate a 99% confidence interval for the variance of the time taken to travel to school for sixth form students from her school. (9)
| Scheme | Marks |
|---|---|
| \(\bar{x} - 2.262\dfrac{s}{\sqrt{10}} = 28.5\) | B1 M1 |
| \(\bar{x} + 2.262\dfrac{s}{\sqrt{10}} = 48.7\) | A1 |
| \(2\bar{x} = 48.7 + 28.5\) or \(2.262\dfrac{s}{\sqrt{10}} = \dfrac{1}{2}(48.7 - 28.5)\) | M1 |
| \(s = 14.1198\ldots\) \((s^2 = 199.36)\) | A1 |
| \(\left\{\dfrac{9\left(14.1198^2\right)}{23.589}, \dfrac{9\left(14.1198^2\right)}{1.735}\right\}\) | M1 B1 B1 |
| \(= (76.0659\ldots, 1034.19\ldots)\) | A1 |
| (9) | |
| (9 marks) |
Notes
B1 awrt 2.262
M1 \(\bar{x} - t\ \textit{value}\dfrac{s}{\sqrt{10}} = 28.5\)
A1 both equations correct
M1 solving simultaneous leading to a value for \(\bar{x}\) or \(s\)
A1 awrt 14.1 or awrt 199
M1 \(\dfrac{9\left(s^2\right)}{\chi^2\,\textit{value}}\)
B1 23.589
B1 1.735
A1 awrt 76.1 and awrt 1030