| Exam Board | WJEC |
|---|---|
| Module | Unit 4 (Unit 4) |
| Session | Specimen |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Validity of normal model |
| Difficulty | Moderate -0.3 This is a straightforward application of normal distribution with standard procedures: justifying normality from a histogram, calculating proportions using z-scores, and conducting a basic one-sample z-test. All parts follow textbook methods with no novel problem-solving required, though it does require understanding of when the normal model is valid and interpreting summary statistics, making it slightly easier than average. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation2.05c Significance levels: one-tail and two-tail2.05d Sample mean as random variable2.05e Hypothesis test for normal mean: known variance |
| Summary statistics | ||||||||||||
| Mean |
| Minimum |
| Median |
| Maximum | ||||||
| 6.89 | 0.296 | 6.45 | 6.63 | 6.88 | 7.08 | 7.48 | ||||||
| Answer | Marks | Guidance |
|---|---|---|
| (a) Bell shaped | B1 | AO2 |
| (b) \(1 - P(6.12 < X < 8.12) = 1 - 0.9949(0744) = 0.0051 \text{ (or } 0.51\%)\) | M1, A1 | AO3, AO1 |
| (c)(i) \(\text{The population of weights of 2p coins is normally distributed. Mean and median in the sample are very similar, suggesting a symmetric distribution.}\) | B1, B1 | AO2, AO2 |
| (c)(ii) \(H_0: \text{The mean weight of all 2p coins in this batch} = 7.12g\) | B1 | AO3 |
| \(H_1: \text{The mean weight of all 2p coins in this batch} < 7.12g \text{ (one-sided)}\) | — | — |
| \(p\text{-value} = P\left(\bar{x} < 6.89 \ | H_0\right) = P\left(z < \frac{6.89-7.12}{0.357/\sqrt{10}}\right)\) | M1 |
| \(= P(z < -3.52(874))\) | A1 | AO1 |
| \(= 0.00021 \text{ (allow } 0.00022)\) | A1 | AO1 |
| \(\text{Since } p\text{-value} < 0.01, \text{ Reject } H_0\) | A1 | AO2 |
| \(\text{Very strong evidence to suggest the mean weight of the batch of 2p coins is less than } 7.12(g)\) | E1 | AO3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(TS = \frac{6.89-7.12}{0.357/\sqrt{10}} = -3.52(874)\) | (M1), (A1) | (AO1), (AO1) |
| \(CV = -2.32(63)\) | (A1), (A1) | (AO1), (AO1) |
| (A1) | (AO2) | |
| \(\text{Very strong evidence to suggest the mean weight of the batch of 2p coins is less than } 7.12(g)\) | (E1) | (AO3) |
**(a)** Bell shaped | B1 | AO2 | Or Most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme; B0 for symmetrical only |
**(b)** $1 - P(6.12 < X < 8.12) = 1 - 0.9949(0744) = 0.0051 \text{ (or } 0.51\%)$ | M1, A1 | AO3, AO1 | Or $P(X < 6.12) + P(X > 8.12)$; M1A0 For 0.9949(0744) |
**(c)(i)** $\text{The population of weights of 2p coins is normally distributed. Mean and median in the sample are very similar, suggesting a symmetric distribution.}$ | B1, B1 | AO2, AO2 | B1B0 The weights of 2p coins are normally distributed. Population must be stated or implied. |
**(c)(ii)** $H_0: \text{The mean weight of all 2p coins in this batch} = 7.12g$ | B1 | AO3 | Or $H_0: \mu = 7.12g$; B0 for $H_1$: Mean $= 7.12g$ |
| $H_1: \text{The mean weight of all 2p coins in this batch} < 7.12g \text{ (one-sided)}$ | — | — | Population must be stated or implied, i.e. the batch of 2p coins |
| $p\text{-value} = P\left(\bar{x} < 6.89 \| H_0\right) = P\left(z < \frac{6.89-7.12}{0.357/\sqrt{10}}\right)$ | M1 | AO1 | |
| $= P(z < -3.52(874))$ | A1 | AO1 | |
| $= 0.00021 \text{ (allow } 0.00022)$ | A1 | AO1 | |
| $\text{Since } p\text{-value} < 0.01, \text{ Reject } H_0$ | A1 | AO2 | FT two-sided test; $p\text{-value} = 2 \times 0.00021 = 0.00042$ |
| $\text{Very strong evidence to suggest the mean weight of the batch of 2p coins is less than } 7.12(g)$ | E1 | AO3 | |
**Alternative Solution:**
| $TS = \frac{6.89-7.12}{0.357/\sqrt{10}} = -3.52(874)$ | (M1), (A1) | (AO1), (AO1) | FT Two-sided test; CVs $= \pm 2.576$ |
| $CV = -2.32(63)$ | (A1), (A1) | (AO1), (AO1) | Since $TS < CV$ Reject $H_0$ |
| (A1) | (AO2) | |
| $\text{Very strong evidence to suggest the mean weight of the batch of 2p coins is less than } 7.12(g)$ | (E1) | (AO3) | |4. Automatic coin counting machines sort, count and batch coins. A particular brand of these machines rejects $2 p$ coins that are less than 6.12 grams or greater than 8.12 grams.
\begin{enumerate}[label=(\alph*)]
\item The histogram represents the distribution of the weight of UK 2p coins supplied by the Royal Mint. This distribution has mean 7.12 grams and standard deviation 0.357 grams.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Weight of UK two pence coins}
\includegraphics[alt={},max width=\textwidth]{b35e94ab-a426-4fca-9ecb-c659e0143ed7-3_602_969_664_589}
\end{center}
\end{figure}
Explain why the weight of 2 p coins can be modelled using a normal distribution.
\item Assume the distribution of the weight of $2 p$ coins is normally distributed. Calculate the proportion of $2 p$ coins that are rejected by this brand of coin counting machine.
\item A manager suspects that a large batch of $2 p$ coins is counterfeit. A random sample of 30 of the suspect coins is selected. Each of the coins in the sample is weighed. The results are shown in the summary statistics table.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | }
\hline
\multicolumn{7}{|c|}{Summary statistics} \\
Mean & \begin{tabular}{ c }
Standard \\
deviation \\
\end{tabular} & Minimum & \begin{tabular}{ c }
Lower \\
quartile \\
\end{tabular} & Median & \begin{tabular}{ c }
Upper \\
quartile \\
\end{tabular} & Maximum \\
\hline
6.89 & 0.296 & 6.45 & 6.63 & 6.88 & 7.08 & 7.48 \\
\hline
\end{tabular}
\end{center}
i) What assumption must be made about the weights of coins in this batch in order to conduct a test of significance on the sample mean? State, with a reason, whether you think this assumption is reasonable.\\
ii) Assuming the population standard deviation is 0.357 grams, test at the $1 \%$ significance level whether the mean weight of the $2 p$ coins in this batch is less than 7.12 grams.
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 4 Q4 [11]}}