| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | T-tests (unknown variance) |
| Type | Single sample t-test |
| Difficulty | Moderate -0.3 This is a straightforward one-sample t-test with all values provided (sample mean, variance estimate, sample size). Students need to state hypotheses, calculate the t-statistic using the standard formula, compare to critical value, and conclude. It's slightly easier than average because it's a direct application of a standard procedure with no complications, though it requires careful execution of the hypothesis testing framework. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| \(H_0: \mu = 200\), \(H_1: \mu > 200\) | B1 | Both hypotheses with \(\mu\) |
| \(t = \dfrac{202-200}{\sqrt{\frac{3.6}{10}}} = \dfrac{10}{3}\) or \(3.3333\ldots\) | M1A1 | M1: allow \(\pm\dfrac{202-200}{\frac{s}{\sqrt{10}}}\); A1: awrt 3.33 |
| Critical value: \(t_9 = (\pm)2.821\) | B1 | Allow \(p\)-value of awrt 0.00438 in place of CV; CV must follow from \(H_1\), sign must match \(t\)-value or be \(\pm\) |
| In critical region, therefore significant evidence to reject \(H_0\) and accept \(H_1\). Significant evidence that the mean weight of the packets of almonds is more than 200 g | A1ft (5) | ft \(t\)-value if B marks awarded; conclusion in context must contain mean weight, almonds or packets and 200g |
## Question 1:
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| $H_0: \mu = 200$, $H_1: \mu > 200$ | B1 | Both hypotheses with $\mu$ |
| $t = \dfrac{202-200}{\sqrt{\frac{3.6}{10}}} = \dfrac{10}{3}$ or $3.3333\ldots$ | M1A1 | M1: allow $\pm\dfrac{202-200}{\frac{s}{\sqrt{10}}}$; A1: awrt 3.33 |
| Critical value: $t_9 = (\pm)2.821$ | B1 | Allow $p$-value of awrt 0.00438 in place of CV; CV must follow from $H_1$, sign must match $t$-value or be $\pm$ |
| In critical region, therefore significant evidence to reject $H_0$ and accept $H_1$. Significant evidence that the **mean weight** of the **packets of almonds** is more than **200 g** | A1ft (5) | ft $t$-value if B marks awarded; conclusion in context must contain **mean weight**, **almonds or packets** and **200g** |
---\begin{enumerate}
\item A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$. To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for $\mu$ and $\sigma ^ { 2 }$ are
\end{enumerate}
$$\bar { x } = 202 \quad \text { and } \quad s ^ { 2 } = 3.6$$
Stating your hypotheses clearly, test, at the $1 \%$ level of significance, whether or not the mean weight of almonds in a packet is more than 200 g .\\
\hfill \mbox{\textit{Edexcel S4 2018 Q1 [5]}}