| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Mixed sum threshold probability |
| Difficulty | Challenging +1.2 This is a standard linear combinations of normal variables question requiring calculation of combined mean/variance and a normal probability, followed by a reverse lookup with inequality reasoning. While it involves multiple steps and careful handling of the 'worst case' scenario in part (b), the techniques are routine for S3 level with no novel insight required beyond recognizing that maximum weight occurs with all men. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)5.04b Linear combinations: of normal distributions |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(M \sim N(80, 100)\), \(W \sim N(69, 25)\); \(X = M_1 + M_2 + M_3 + M_4 + M_5 + M_6 + W_1 + W_2 + W_3\) | ||
| \(X \sim N(687, 675)\) | M1 A1 | B1 for mean 687; B1 for correct variance (675) or sd (\(15\sqrt{3}\)) |
| \(P(X > 700) = P\left(Z > \frac{700-687}{\sqrt{675}}\right) = P(Z > 0.500...)\) | M1 | for standardising with 700, 687 and their standard deviation |
| \(= 1 - 0.6915 = 0.3085\) (Calculator gives 0.3084) | A1 | for answer between 0.308 – 0.309 |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Let \(Y\) = Number of men in the lift; \(Y \sim N(80x, 100x)\) | M1 | for setting up normal distribution with mean \(80x\) and variance \(100x\); sd \(= 10\sqrt{x}\) |
| \(P(Y > 700) = P\left(Z > \frac{700-80x}{10\sqrt{x}}\right) < 0.025\) | M1 | for standardising with 700, their mean and their standard deviation |
| \(\frac{700-80x}{10\sqrt{x}} > 1.96\) | B1 | for equation or inequality set to 1.96 (Allow –1.96) |
| \(80x + 19.6\sqrt{x} - 700 [< 0]\) or \(6400x^2 - 112384.16x + 490000 [> 0]\) | M1 | for a correct 3TQ ft their mean and standard deviation |
| Solving leading to \(\sqrt{x} < 2.838...\) or \(x < 8.05...\) | M1 | for attempt to solve 3TQ with \(\sqrt{x} < ...\) or \(x < ...\); allow \(=\) or \(\geq\); if answer incorrect must see quadratic formula/completing the square |
| So \(c = 8\) (people) | A1 | cao |
| (6) |
## Question 6:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $M \sim N(80, 100)$, $W \sim N(69, 25)$; $X = M_1 + M_2 + M_3 + M_4 + M_5 + M_6 + W_1 + W_2 + W_3$ | | |
| $X \sim N(687, 675)$ | M1 A1 | B1 for mean 687; B1 for correct variance (675) or sd ($15\sqrt{3}$) |
| $P(X > 700) = P\left(Z > \frac{700-687}{\sqrt{675}}\right) = P(Z > 0.500...)$ | M1 | for standardising with 700, 687 and their standard deviation |
| $= 1 - 0.6915 = 0.3085$ (Calculator gives 0.3084) | A1 | for answer between 0.308 – 0.309 |
| | **(4)** | |
### Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Let $Y$ = Number of men in the lift; $Y \sim N(80x, 100x)$ | M1 | for setting up normal distribution with mean $80x$ and variance $100x$; sd $= 10\sqrt{x}$ |
| $P(Y > 700) = P\left(Z > \frac{700-80x}{10\sqrt{x}}\right) < 0.025$ | M1 | for standardising with 700, their mean and their standard deviation |
| $\frac{700-80x}{10\sqrt{x}} > 1.96$ | B1 | for equation or inequality set to 1.96 (Allow –1.96) |
| $80x + 19.6\sqrt{x} - 700 [< 0]$ or $6400x^2 - 112384.16x + 490000 [> 0]$ | M1 | for a correct 3TQ ft their mean and standard deviation |
| Solving leading to $\sqrt{x} < 2.838...$ or $x < 8.05...$ | M1 | for attempt to solve 3TQ with $\sqrt{x} < ...$ or $x < ...$; allow $=$ or $\geq$; if answer incorrect must see quadratic formula/completing the square |
| So $c = 8$ (people) | A1 | cao |
| | **(6)** | |
---6 A particular lift has a maximum load capacity of 700 kg .\\
The weights of men are normally distributed with mean 80 kg and standard deviation 10 kg .
The weights of women are normally distributed with mean 69 kg and standard deviation 5 kg .
You may assume that weights of people are independent.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that when 6 men and 3 women are in the lift, the load exceeds 700 kg .
A sign in the lift states: "Maximum number of people in the lift is $c$ "
\item Find the value of $c$ such that the probability of the load exceeding 700 kg is less than $2.5 \%$ no matter the gender of the occupants.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 2022 Q6 [10]}}