| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Z-tests (known variance) |
| Type | Two-tail z-test |
| Difficulty | Standard +0.3 This is a straightforward S3 hypothesis testing question with standard parts: stating sampling distribution, calculating a probability using normal tables, and conducting a two-tailed z-test with known variance. All steps are routine applications of formulas with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.05a Sample mean distribution: central limit theorem5.05c Hypothesis test: normal distribution for population mean |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Let \(W\) = weight of component \(\therefore W \sim N(46.7, 1.8)\). \(\overline{W} \sim N(46.7, \frac{1.8^2}{13}) = N(46.7, 0.15)\) | M1 A1 | |
| (b) \(P(\overline{W} > 47) = P(Z > \frac{47-46.7}{\sqrt{0.15}})\) | M1 | |
| \(= P(Z > 0.77) = 1 - 0.7794 = 0.2206\) | M1 A1 | |
| (c) \(H_0: \mu = 46.7\); \(H_1: \mu \neq 46.7\). 5% level \(\therefore\) C.R. is \(z < -1.96\) or \(z > 1.96\). Test statistic \(= \frac{46.5-46.7}{\sqrt{\frac{1.8}{13}}} = -0.816\) | B1, B1, M2 A2 | |
| Not in C.R. do not reject \(H_0\). No evidence of change in mean weight | A1 | Total: 12 marks |
(a) Let $W$ = weight of component $\therefore W \sim N(46.7, 1.8)$. $\overline{W} \sim N(46.7, \frac{1.8^2}{13}) = N(46.7, 0.15)$ | M1 A1 |
(b) $P(\overline{W} > 47) = P(Z > \frac{47-46.7}{\sqrt{0.15}})$ | M1 |
$= P(Z > 0.77) = 1 - 0.7794 = 0.2206$ | M1 A1 |
(c) $H_0: \mu = 46.7$; $H_1: \mu \neq 46.7$. 5% level $\therefore$ C.R. is $z < -1.96$ or $z > 1.96$. Test statistic $= \frac{46.5-46.7}{\sqrt{\frac{1.8}{13}}} = -0.816$ | B1, B1, M2 A2 |
Not in C.R. do not reject $H_0$. No evidence of change in mean weight | A1 | Total: 12 marks6. The weight of a particular electrical component is normally distributed with a mean of 46.7 grams and a variance of 1.8 grams $^ { 2 }$. The component is sold in boxes of 12 .
\begin{enumerate}[label=(\alph*)]
\item State the distribution of the mean weight of the components in one box.
\item Find the probability that the mean weight of the components in a randomly chosen box is more than 47 grams.\\
(3 marks)\\
After a break in production the component manufacturer wishes to find out if the mean weight of the components has changed. A random sample of 30 components is found to have a mean weight of 46.5 grams.
\item Assuming that the variance of the weight of the components is unchanged, test at the $5 \%$ level of significance if there has been any change in the mean weight of the components.\\
(7 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q6 [12]}}