| Exam Board | Edexcel |
|---|---|
| Module | FS2 (Further Statistics 2) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Known variance (z-distribution) |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question requiring standard procedures: calculating a confidence interval using known variance (z-distribution), finding a critical region for variance (chi-squared), and interpreting results. All techniques are routine for Further Statistics 2, with no novel problem-solving required beyond applying memorized formulas and comparing values to critical regions. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1.6449\) | B1 | For realising a normal distribution must be used; correct value 1.6449 or better |
| \(30.03 \pm \frac{0.868}{\sqrt{15}} \times 1.6449\) | M1 | For \(30.03 \pm \frac{0.868}{\sqrt{15}} \times z\); may be implied by a correct CI |
| \((29.6613\ldots,\ 30.3986\ldots)\) | A1 | awrt 29.7 and 30.4 from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\chi^2_{14}(0.1)\ \text{CV} = 21.064\) | B1 | For realising chi squared distribution must be used; CV = awrt 21.06 |
| Reject \(H_0\) if \(\frac{14S^2}{0.868^2} > 21.06\) | M1 | Correct method comparing \(\frac{14S^2}{0.868^2}\) to \(19 < \chi^2_{14} < 24\) (condone equals instead of \(>\)) |
| Critical region is \(S^2 > 1.133\ldots\) | A1 | Correct CR, allow awrt 1.13 only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Insufficient evidence that the machine is not working properly as 30.06 is within the confidence interval | M1 | Dependent on answer to (a) or (b); correct inference with one correct comparison; must mention machine at least once |
| and \(1.02^2\ (1.0404)\) is not in the CR | A1ft | Dependent on 30.06 in CI and \(1.02^2\) not in CR; both correct comparisons, no contradictory statements; do not accept comparison of 30.06 with just one limit of CI |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1.6449$ | B1 | For realising a normal distribution must be used; correct value 1.6449 or better |
| $30.03 \pm \frac{0.868}{\sqrt{15}} \times 1.6449$ | M1 | For $30.03 \pm \frac{0.868}{\sqrt{15}} \times z$; may be implied by a correct CI |
| $(29.6613\ldots,\ 30.3986\ldots)$ | A1 | awrt 29.7 and 30.4 from correct working |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\chi^2_{14}(0.1)\ \text{CV} = 21.064$ | B1 | For realising chi squared distribution must be used; CV = awrt 21.06 |
| Reject $H_0$ if $\frac{14S^2}{0.868^2} > 21.06$ | M1 | Correct method comparing $\frac{14S^2}{0.868^2}$ to $19 < \chi^2_{14} < 24$ (condone equals instead of $>$) |
| Critical region is $S^2 > 1.133\ldots$ | A1 | Correct CR, allow awrt 1.13 only |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Insufficient evidence that the machine is not working properly as 30.06 is within the confidence interval | M1 | Dependent on answer to (a) or (b); correct inference with one correct comparison; must mention machine at least once |
| and $1.02^2\ (1.0404)$ is not in the CR | A1ft | Dependent on 30.06 in CI **and** $1.02^2$ not in CR; both correct comparisons, no contradictory statements; do not accept comparison of 30.06 with just one limit of CI |
---\begin{enumerate}
\item A factory produces bolts. The lengths of the bolts are normally distributed with mean $\mu \mathrm { mm }$ and standard deviation 0.868 mm
\end{enumerate}
A random sample of 15 of these bolts is taken and the mean length is 30.03 mm\\
(a) Calculate a 90\% confidence interval for $\mu$
A suitable test, at the $10 \%$ level of significance, is carried out using these 15 bolts, to see whether or not there is evidence that the variance of the length of the bolts has increased.\\
(b) Calculate the critical region for $S ^ { 2 }$
The manager of the factory decides that, in future, he will check each month whether the machine making the bolts is working properly. He uses a $10 \%$ level of significance to test whether or not there is evidence that
\begin{itemize}
\item the mean length of the bolts has changed
\item the variance of the length of the bolts has increased
\end{itemize}
The next month a random sample of 15 bolts is taken.\\
The mean length of these bolts is 30.06 mm and the standard deviation is 1.02 mm\\
(c) With reference to your answers to part (a) and part (b), state whether or not there is any evidence that the machine is not working properly.\\
Give reasons for your answer.
\hfill \mbox{\textit{Edexcel FS2 2024 Q3 [8]}}