| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2023 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Unbiased estimates calculation |
| Difficulty | Standard +0.3 This is a straightforward one-sample t-test with clear hypotheses and small sample size. Part (a) requires standard calculations of sample mean and unbiased variance (dividing by n-1), part (b) is a routine one-tailed t-test at 5% level, and part (c) asks for a practical consideration. The question is slightly above average difficulty only because it's Further Maths content and requires knowledge of the t-distribution, but the execution is mechanical with no conceptual challenges or novel problem-solving required. |
| Spec | 5.05b Unbiased estimates: of population mean and variance5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\sum x = 2759\), \(\sum x^2 = 846081\) | B1 |
| \(\bar{\mu} = 306.555...\) | M1A1 | M1 for appropriate use of calculator or use of \(\bar{\sigma}^2 = \frac{1}{n-1}(\sum x^2 - n\bar{x}^2)\). Allow 33.7122 from rounding \(\bar{\mu}\) to 306.56. M1A0 for 40.6096... from \(\bar{x} = 306.55\). M1A0 for 6.12 from \(\bar{x} = 306.6\). FT their \(\bar{\mu}\) for M1 only, provided \(\bar{\sigma}^2 > 0\) |
| \(\hat{\sigma}^2 = \frac{1}{8}(846081 - 9 \times 306.555...^2) = \frac{331}{9} = 36.777...\) | ||
| (b) | \(H_0: \mu = 305\) and \(H_1: \mu > 305\) | B1 |
| \(DF = 8\) | B1 | si |
| \(CV = 1.860\) | B1 | FT their DF |
| \(t = \frac{306.5555...-305}{\sqrt{\frac{36.7777...}{9}}}\) | M1 | FT their \(\bar{\mu}\) and \(\hat{\sigma}^2\) |
| \(t = 0.7695...\) | A1 | cao. Accept 0.77 from correct working. Allow 0.806 from \(\bar{\mu} = 306.56\) and \(\hat{\sigma}^2 = 33.71(22)\) |
| Since \(0.7695 < 1.860\) there is insufficient evidence to reject \(H_0\). | m1 | FT their \(t\). Dep on use of t-distribution. |
| There is insufficient evidence to say that this is an old kettle. | A1 | cso |
| (c) | Valid factor. e.g. the initial water temperature. e.g. the initial kettle temperature. e.g. the ambient temperature. e.g. the volume of water. e.g. the voltage going to the kettle. e.g. the mineral content of the water | E1 |
| Total [11] |
(a) | $\sum x = 2759$, $\sum x^2 = 846081$ | B1 | At least 1dp |
| $\bar{\mu} = 306.555...$ | M1A1 | M1 for appropriate use of calculator or use of $\bar{\sigma}^2 = \frac{1}{n-1}(\sum x^2 - n\bar{x}^2)$. Allow 33.7122 from rounding $\bar{\mu}$ to 306.56. M1A0 for 40.6096... from $\bar{x} = 306.55$. M1A0 for 6.12 from $\bar{x} = 306.6$. FT their $\bar{\mu}$ for M1 only, provided $\bar{\sigma}^2 > 0$ |
| $\hat{\sigma}^2 = \frac{1}{8}(846081 - 9 \times 306.555...^2) = \frac{331}{9} = 36.777...$ | |
(b) | $H_0: \mu = 305$ and $H_1: \mu > 305$ | B1 | |
| $DF = 8$ | B1 | si |
| $CV = 1.860$ | B1 | FT their DF |
| $t = \frac{306.5555...-305}{\sqrt{\frac{36.7777...}{9}}}$ | M1 | FT their $\bar{\mu}$ and $\hat{\sigma}^2$ |
| $t = 0.7695...$ | A1 | cao. Accept 0.77 from correct working. Allow 0.806 from $\bar{\mu} = 306.56$ and $\hat{\sigma}^2 = 33.71(22)$ |
| Since $0.7695 < 1.860$ there is insufficient evidence to reject $H_0$. | m1 | FT their $t$. Dep on use of t-distribution. |
| There is insufficient evidence to say that this is an old kettle. | A1 | cso |
(c) | Valid factor. e.g. the initial water temperature. e.g. the initial kettle temperature. e.g. the ambient temperature. e.g. the volume of water. e.g. the voltage going to the kettle. e.g. the mineral content of the water | E1 | |
| **Total [11]** | | |
---\begin{enumerate}
\item The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.\\
305\\
295\\
310\\
310\\
315\\
307\\
300\\
311\\
306
\end{enumerate}
You may assume the times taken to boil the full kettle are normally distributed.\\
(a) Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.\\
(b) Test, at the $5 \%$ level of significance, whether there is evidence to suggest that this is an old kettle.\\
(c) State a factor that Alun should control when carrying out this investigation.\\
\hfill \mbox{\textit{WJEC Further Unit 5 2023 Q1 [11]}}