| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | F-test and chi-squared for variance |
| Type | Pooled variance from two samples |
| Difficulty | Standard +0.8 This S4 question requires knowledge of pooling sample variances from independent samples and constructing chi-squared confidence intervals for variance. While the individual techniques are standard, combining them correctly (pooling with appropriate degrees of freedom, then applying chi-squared distribution) requires careful multi-step reasoning beyond routine S2 content, making it moderately challenging for Further Maths statistics. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(s_p^2 = \frac{6s_x^2 + 3s_y^2}{9}\) (= 192.03…) | M1 | For attempting \(s_p^2\) |
| \(1.735 < \frac{9s_p^2}{\sigma^2} < 23.589\) | B1M1B1 | 1st B1 for 1.735 (or better); 2nd M1 for use of \(\frac{9s_p^2}{\sigma^2}\), follow through their \(s_p^2\); 2nd B1 for 23.589 (or better) |
| 99% confidence interval is (73.26…, 996.14…) | A1 | awrt (73.3, 996); both values correct to awrt 3 sf |
| Total: 5 marks |
# Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $s_p^2 = \frac{6s_x^2 + 3s_y^2}{9}$ (= 192.03…) | M1 | For attempting $s_p^2$ |
| $1.735 < \frac{9s_p^2}{\sigma^2} < 23.589$ | B1M1B1 | 1st B1 for 1.735 (or better); 2nd M1 for use of $\frac{9s_p^2}{\sigma^2}$, follow through their $s_p^2$; 2nd B1 for 23.589 (or better) |
| 99% confidence interval is (73.26…, 996.14…) | A1 | awrt **(73.3, 996)**; both values correct to awrt 3 sf |
| **Total: 5 marks** | | |
---\begin{enumerate}
\item Two independent random samples $X _ { 1 } , X _ { 2 } , \ldots , X _ { 7 }$ and $Y _ { 1 } , Y _ { 2 } , Y _ { 3 } , Y _ { 4 }$ were taken from different normal populations with a common standard deviation $\sigma$. The following sample statistics were calculated.
\end{enumerate}
$$s _ { x } = 14.67 \quad s _ { y } = 12.07$$
Find the $99 \%$ confidence interval for $\sigma ^ { 2 }$ based on these two samples.\\
\hfill \mbox{\textit{Edexcel S4 2011 Q2 [5]}}