| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Confidence intervals |
| Type | Unbiased estimates then CI |
| Difficulty | Moderate -0.3 This is a straightforward S3 confidence interval question requiring standard formulas (unbiased estimates, z-interval with known variance) and routine interpretation. The calculations are direct with no conceptual challenges, making it slightly easier than average but still requiring proper statistical understanding. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} = \frac{60}{15} = 4\) | B1 | 4 cao |
| \(s^2 = \frac{1}{14}(1946 - 15 \times 4^2) = 121.857\ldots\) | M1,A1 | M1 use of complete, correct formula and attempt to substitute; A1 awrt 122 or \(\frac{853}{7}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\bar{x} \pm 1.96 \times \frac{10}{\sqrt{15}} = 4 \pm 5.06\) | M1,A1 | Accept use of \(\bar{x} \pm z \times \frac{10 \text{ or "their } s\text{"}}{\sqrt{15}}\); A1 all correct; accept \(\bar{x} = 0835\) |
| \((-1.06, 9.06)\) | A1 | Can be implied from correct interval below |
| \((0829:56, 0840:04)\) | A1 | Accept \((0829.94, 0840.06)\) or expressed using words or as an inequality; accept answers to nearest minute i.e. \((0830, 0840)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Paul samples times of buses randomly or independently of each other | B1 | Context required |
| Answer | Marks | Guidance |
|---|---|---|
| \(0/0831/8.31\text{(am)}\) is 'contained in' the confidence interval | 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' |
| Paul's belief is not supported / 0831 arrival time is reasonable | A1cao | Must contain some context |
## Question 5:
### Part (a):
$\bar{x} = \frac{60}{15} = 4$ | B1 | 4 cao
$s^2 = \frac{1}{14}(1946 - 15 \times 4^2) = 121.857\ldots$ | M1,A1 | M1 use of complete, correct formula and attempt to substitute; A1 awrt 122 or $\frac{853}{7}$
### Part (b)(i):
$\bar{x} \pm 1.96 \times \frac{10}{\sqrt{15}} = 4 \pm 5.06$ | M1,A1 | Accept use of $\bar{x} \pm z \times \frac{10 \text{ or "their } s\text{"}}{\sqrt{15}}$; A1 all correct; accept $\bar{x} = 0835$
$(-1.06, 9.06)$ | A1 | Can be implied from correct interval below
$(0829:56, 0840:04)$ | A1 | Accept $(0829.94, 0840.06)$ or expressed using words or as an inequality; accept answers to nearest minute i.e. $(0830, 0840)$
### Part (b)(ii):
Paul samples times of **buses randomly** or **independently** of each other | B1 | Context required
### Part (c):
$0/0831/8.31\text{(am)}$ is 'contained in' the confidence interval | 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'
Paul's belief is not supported / 0831 arrival time is reasonable | A1cao | Must contain some context
---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
$$X = \text { number of minutes after } 0831 \text { when the bus arrives. }$$
His results are summarised below.
$$n = 15 \quad \sum x = 60 \quad \sum x ^ { 2 } = 1946$$
\begin{enumerate}[label=(\alph*)]
\item Calculate unbiased estimates of the mean, $\mu$, and the variance of $X$.
Using the mean of Paul's sample and given $X \sim \mathrm {~N} \left( \mu , 10 ^ { 2 } \right)$
\item \begin{enumerate}[label=(\roman*)]
\item calculate a 95\% confidence interval for the mean arrival time at work for this company bus.
\item State an assumption you made about the values in the sample obtained by Paul.
\end{enumerate}\item Comment on Paul's belief. Justify your answer.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2017 Q5 [10]}}