| Exam Board | Edexcel |
|---|---|
| Module | S3 (Statistics 3) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear combinations of normal random variables |
| Type | Two or more different variables |
| Difficulty | Standard +0.3 This is a standard S3 question on linear combinations of independent normal variables. Part (a) requires forming X+Y and standardizing; part (b) requires forming Y-2X and standardizing. Both are direct applications of the formula for combining normals with no conceptual surprises, making it slightly easier than average for A-level. |
| 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 distributions |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Let \(F\) = time on French and \(E\) = time on English. Let \(A = F + E \therefore A \sim N(55 + 90, 10^2 + 18^2) = N(145, 424)\) | M1 A1 | |
| \(P(A > 120) = P(Z > \frac{120-145}{\sqrt{424}})\) | M1 | |
| \(= P(Z > -1.21) = 0.8869\) | M1 A1 | |
| (b) \(P(E > 2F) = P(E - 2F > 0)\). Let \(B = E - 2F \therefore B \sim N(90 - 2 \times 55, 18^2 + 4 \times 10^2) = N(-20, 724)\) | M1, M1 A1 | |
| \(P(B > 0) = P(Z > \frac{0+20}{\sqrt{724}})\) | M1 | |
| \(= P(Z > 0.74) = 1 - 0.7704 = 0.2296\) | M1 A1 | Total: 11 marks |
(a) Let $F$ = time on French and $E$ = time on English. Let $A = F + E \therefore A \sim N(55 + 90, 10^2 + 18^2) = N(145, 424)$ | M1 A1 |
$P(A > 120) = P(Z > \frac{120-145}{\sqrt{424}})$ | M1 |
$= P(Z > -1.21) = 0.8869$ | M1 A1 |
(b) $P(E > 2F) = P(E - 2F > 0)$. Let $B = E - 2F \therefore B \sim N(90 - 2 \times 55, 18^2 + 4 \times 10^2) = N(-20, 724)$ | M1, M1 A1 |
$P(B > 0) = P(Z > \frac{0+20}{\sqrt{724}})$ | M1 |
$= P(Z > 0.74) = 1 - 0.7704 = 0.2296$ | M1 A1 | Total: 11 marks3. The time that a school pupil spends on French homework each week is normally distributed with a mean of 55 minutes and a standard deviation of 10 minutes.
The time that this pupil spends on English homework each week is normally distributed with a mean of 1 hour 30 minutes and a standard deviation of 18 minutes.
Find the probability that in a randomly chosen week
\begin{enumerate}[label=(\alph*)]
\item the pupil spends more than 2 hours in total doing French and English homework,
\item the pupil spends more than twice as long doing English homework as he spends doing French homework.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel S3 Q3 [11]}}