S3 June 2017 Q5
5. Paul takes the company bus to work. According to the bus timetable he should arrive at work at 0831. Paul believes the bus is not reliable and often arrives late. Paul decides to test the arrival time of the bus and carries out a survey. He records the values of the random variable
\[X = \text{number of minutes after 0831 when the bus arrives.}\]His results are summarised below.
\[n = 15 \qquad \sum x = 60 \qquad \sum x^2 = 1946\]Using the mean of Paul’s sample and given \(X \sim \mathrm{N}(\mu, 10^2)\)
| Scheme | Marks |
|---|---|
| \(\bar{x} = \dfrac{60}{15} = 4\) | B1 |
| \(s^2 = \dfrac{1}{14}\left(1946 - 15 \times 4^2\right) = 121.857\ldots\) | M1,A1 |
| (3) |
Notes
B1 4 cao
M1 Use of complete, correct formula and attempt to substitute.
A1 awrt 122 or \(\dfrac{853}{7}\)
| Scheme | Marks |
|---|---|
| (i) \(\bar{x} \pm 1.96 \times \dfrac{10}{\sqrt{15}} = 4 \pm 5.06\) | M1,A1 |
| \((-1.06,\ 9.06)\) | A1 |
| \((082956,\ 084004)\) | A1 |
| (ii) Paul samples times of buses randomly or independently of each other | B1 |
| (5) |
Notes
M1 Accept use of \(\bar{x} \pm z \times \dfrac{10 \text{ or "their } s\text{"}}{\sqrt{15}}\),
A1 all correct.
Accept \(\bar{x} = 0835\).
A1 Can be implied from correct interval below.
A1 Accept \((0829.94,\ 0840.06)\) or expressed using words or as an inequality.
Accept answers to the nearest minute ie (0830,0840).
B1 (ii) Context required.
| Scheme | Marks |
|---|---|
| 0 / 0831 / 8.31(am) is ‘contained in’ the confidence interval | M1 |
| Paul’s belief is not supported / 0831 arrival time is reasonable | A1cao |
| (2) | |
| (10 marks) |
Notes
M1 Award if comment about their interval is correct. Only accept ‘above the lower limit of’ etc if the statement taken as a whole clearly means ‘contained in’.
A1cao Must contain some context