| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2021 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | One-tail z-test (lower tail) |
| Difficulty | Moderate -0.3 This is a straightforward one-sample z-test with all information provided directly: known variance, clear hypotheses, and standard 5% significance level. Students must set up H₀: μ=30 vs H₁: μ<30, calculate z = (29.5-30)/(2.5/√80) ≈ -1.79, and compare to critical value -1.645. While it requires proper hypothesis test structure, it's a textbook application with no conceptual challenges or multi-step reasoning, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation2.05e Hypothesis test for normal mean: known variance |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance Notes |
| \(H_0: \mu = 30 \quad H_1: \mu < 30\) | B1 | Both hypotheses correct in terms of \(\mu\) |
| \(z = \dfrac{29.5 - 30}{\dfrac{2.5}{\sqrt{80}}}\) | M1 | For attempting test statistic, allow \(\pm\). Condone \(\sqrt{\dfrac{2.5}{80}}\) |
| \(z = -1.7888\ldots\) awrt \(-1.79\) | A1 | Allow \(\ |
| \(-1.7888 < -1.6449\) | B1 | \(\ |
| Reject \(H_0\) or significant result or in the critical region. There is evidence to support the manager's claim. | A1 | For correct conclusion. Allow the manager's claim in words if it includes screws and less (oe) |
| Total | 5 |
## Question 1:
| Answer/Working | Mark | Guidance Notes |
|---|---|---|
| $H_0: \mu = 30 \quad H_1: \mu < 30$ | B1 | Both hypotheses correct in terms of $\mu$ |
| $z = \dfrac{29.5 - 30}{\dfrac{2.5}{\sqrt{80}}}$ | M1 | For attempting test statistic, allow $\pm$. Condone $\sqrt{\dfrac{2.5}{80}}$ |
| $z = -1.7888\ldots$ awrt $-1.79$ | A1 | Allow $\|z\| = 1.7888\ldots$ Allow $p$ value of $0.0367$ or awrt $0.0368$ or $CR \leqslant 29.54$ |
| $-1.7888 < -1.6449$ | B1 | $\|CV\| = 1.6449$ or better (ignore any comparisons). Allow $CR \leqslant 29.54$. SC: If $p$ value of $0.0367$ or awrt $0.0368$, award B1 if 2nd A1 is awarded |
| Reject $H_0$ or significant result or in the critical region. There is evidence to support the manager's claim. | A1 | For correct conclusion. Allow the manager's claim in words if it includes screws and less (oe) |
| **Total** | **5** | |\begin{enumerate}
\item A machine makes screws with a mean length of 30 mm and a standard deviation of 2.5 mm .
\end{enumerate}
A manager claims that, following some repairs, the machine is now making screws with a mean length of less than 30 mm . The manager takes a random sample of 80 screws and finds that they have a mean length of 29.5 mm .
Use a suitable test, at the $5 \%$ level of significance, to determine whether there is evidence to support the manager's claim. State your hypotheses clearly.
\hfill \mbox{\textit{Edexcel S3 2021 Q1 [5]}}