| Exam Board | Edexcel |
|---|---|
| Module | S4 (Statistics 4) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (lower tail) |
| Difficulty | Standard +0.3 This is a straightforward one-tail z-test with given summary statistics. Students must calculate sample mean and variance, perform a standard hypothesis test procedure, and compare to critical value. While it requires multiple steps (calculate statistics, set up hypotheses, find test statistic, conclude), each step follows a standard algorithm taught in S4 with no novel problem-solving required. Slightly above average difficulty due to being Further Maths content and requiring careful execution of the full hypothesis testing framework. |
| Spec | 5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(H_0: \mu = 5\); \(H_1: \mu < 5\) | B1 | both |
| CR: \(t_9(0.01) > 2.821\) | B1 | |
| \(\bar{x} = 4.91\) | B1 | |
| \(s^2 = \frac{1}{9}\left(241.2 - \frac{49.1^2}{10}\right) = 0.0132222\) | M1 A1 | s = awrt 0.115 |
| \(t = \frac{ | 4.91 - 5 | }{\frac{\sqrt{0.013222}}{\sqrt{10}}} = \pm 2.475\) |
| Since 2.475 is not in the critical region there is insufficient evidence to reject \(H_0\) and conclude that the mean diameter of the bolts is not less than (not equal to) 5mm. | A1ft |
# Question 1:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu = 5$; $H_1: \mu < 5$ | B1 | both |
| CR: $t_9(0.01) > 2.821$ | B1 | |
| $\bar{x} = 4.91$ | B1 | |
| $s^2 = \frac{1}{9}\left(241.2 - \frac{49.1^2}{10}\right) = 0.0132222$ | M1 A1 | s = awrt 0.115 |
| $t = \frac{|4.91 - 5|}{\frac{\sqrt{0.013222}}{\sqrt{10}}} = \pm 2.475$ | M1 A1 | 2.47 – 2.48 |
| Since 2.475 is not in the critical region there is insufficient evidence to reject $H_0$ and conclude that the mean diameter of the bolts is not less than (not equal to) 5mm. | A1ft | |
---\begin{enumerate}
\item A company manufactures bolts with a mean diameter of 5 mm . The company wishes to check that the diameter of the bolts has not decreased. A random sample of 10 bolts is taken and the diameters, $x \mathrm {~mm}$, of the bolts are measured. The results are summarised below.
\end{enumerate}
$$\sum x = 49.1 \quad \sum x ^ { 2 } = 241.2$$
Using a $1 \%$ level of significance, test whether or not the mean diameter of the bolts is less than 5 mm .\\
(You may assume that the diameter of the bolts follows a normal distribution.)\\
\hfill \mbox{\textit{Edexcel S4 2009 Q1 [8]}}