| Exam Board | WJEC |
|---|---|
| Module | Further Unit 5 (Further Unit 5) |
| Year | 2023 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Sample size determination |
| Difficulty | Standard +0.3 This is a straightforward application of sampling distributions and linear combinations of normal variables. Parts (a) and (b) are standard textbook exercises on sampling distributions of means, (c) requires combining independent normals (a routine Further Maths technique), and (d) asks for a basic modelling assumption. All parts follow well-established procedures with no novel insight required, making it slightly easier than average for Further Maths content. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.04a Linear combinations: E(aX+bY), Var(aX+bY)5.04b Linear combinations: of normal distributions5.05a Sample mean distribution: central limit theorem5.05d Confidence intervals: using normal distribution |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\bar{X} \sim N\left(75, \frac{10^2}{5}\right)\) | B1 |
| \(P(\bar{X} < 70) = 0.13177...\) | M1A1 | |
| ALTERNATIVE METHOD | ||
| \(T = X_1 + X_2 + ... + X_5\) | ||
| \(T \sim N(375, 500)\) | (B1) | |
| \(P(T < 350) = 0.13177...\) | (M1A1) | |
| (b) | \(\bar{X} \sim N\left(75, \frac{10^2}{n}\right)\) | B1 |
| \(P(\bar{X} > 80) \approx 0.007\) | ||
| \(P\left(z > \frac{80 - 75}{\sqrt{\frac{100}{n}}}\right) \approx 0.007\) | M1 | Standardising accept (75 – 80) for numerator |
| \(\frac{80 - 75}{\sqrt{\frac{100}{n}}} \approx 2.4572\) | M1B1 | M1 for correct standardisation set equal to \(2 \leq k \leq 3\). B1 for 2.457 or better |
| \(n = 24\) | A1 | cao |
| (c) | Let \(T = X_1 + X_2 + X_3 + Y_1 + Y_2 + Y_3 + Y_4\) | |
| \(E(T) = 497\) | B1 M1 A1 | |
| \(\text{Var}(T) = 3 \times 100 + 4 \times 36\) | ||
| \(\text{Var}(T) = 444\) | ||
| \(P(T > 500) = 0.44339...\) | B1 | From tables 0.44433 |
| (d) | Valid assumption. e.g. the workers do not carry any extra baggage. e.g. mass of workers' clothes may be ignored. | E1 |
| Total [13] |
(a) | $\bar{X} \sim N\left(75, \frac{10^2}{5}\right)$ | B1 | si oe |
| $P(\bar{X} < 70) = 0.13177...$ | M1A1 | |
| ALTERNATIVE METHOD | | |
| $T = X_1 + X_2 + ... + X_5$ | | |
| $T \sim N(375, 500)$ | (B1) | |
| $P(T < 350) = 0.13177...$ | (M1A1) | |
(b) | $\bar{X} \sim N\left(75, \frac{10^2}{n}\right)$ | B1 | si |
| $P(\bar{X} > 80) \approx 0.007$ | | |
| $P\left(z > \frac{80 - 75}{\sqrt{\frac{100}{n}}}\right) \approx 0.007$ | M1 | Standardising accept (75 – 80) for numerator |
| $\frac{80 - 75}{\sqrt{\frac{100}{n}}} \approx 2.4572$ | M1B1 | M1 for correct standardisation set equal to $2 \leq k \leq 3$. B1 for 2.457 or better |
| $n = 24$ | A1 | cao |
(c) | Let $T = X_1 + X_2 + X_3 + Y_1 + Y_2 + Y_3 + Y_4$ | | |
| $E(T) = 497$ | B1 M1 A1 | |
| $\text{Var}(T) = 3 \times 100 + 4 \times 36$ | | |
| $\text{Var}(T) = 444$ | | |
| $P(T > 500) = 0.44339...$ | B1 | From tables 0.44433 |
(d) | Valid assumption. e.g. the workers do not carry any extra baggage. e.g. mass of workers' clothes may be ignored. | E1 | |
| **Total [13]** | | |
---5. The masses, $X$, in kg, of men who work for a large company are normally distributed with mean 75 and standard deviation 10.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that the mean mass of a random sample of 5 men is less than 70 kg .
\item The mean mass, in kg , of a random sample of $n$ men drawn from this distribution is $\bar { X }$. Given that $\mathrm { P } ( \bar { X } > 80 )$ is approximately $0 \cdot 007$, find $n$.
The masses, in kg, of women who work for the company are normally distributed with mean 68 and standard deviation 6 . A lift in the company building will not move if the total mass in the lift is more than 500 kg .
\item A random sample of 3 men and 4 women get in the lift. Find the probability that the lift will not move.
\item State a modelling assumption you have made in calculating your answer for part (c).
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 5 2023 Q5 [13]}}