| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Moderate -0.8 This is a straightforward normal distribution question requiring only standard z-score calculations and inverse normal lookups. All parts are routine applications of basic techniques with no problem-solving insight needed, making it easier than average but not trivial since it requires correct use of tables/calculator. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(X > 43.2) = P\!\left(Z > \dfrac{43.2 - 41.2}{3.6}\right) = P(Z > 0.5556)\) | M1 | Use of \(\pm\)standardisation formula once, allow continuity correction, not \(\sigma^2\), \(\sqrt{\sigma}\). |
| \(1 - \Phi(0.5556) = 1 - 0.7108\) | M1 | Appropriate area \(\Phi\), from final process, must be probability. |
| \(0.289\) | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Probability \(= 1 - \textit{their}\ \textbf{(a)} = 1 - 0.2892 = 0.7108\) | B1FT | \(1 - \textit{their}\ \textbf{(a)}\) or correct. |
| \(0.7108 \times 365 = 259.4\); answer: \(259, 260\) | B1FT | FT *their* 4SF (or better) probability, final answer must be positive integer. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(z = \pm 1.645\) | B1 | CAO, critical \(z\) value. |
| \(\dfrac{t - 41.2}{3.6} = -1.645\) | M1 | Use of \(\pm\)standardisation formula with \(\mu\), \(\sigma\) equated to a \(z\)-value, no continuity correction, allow \(\sigma^2\), \(\sqrt{\sigma}\). |
| \(t = 35.3\) | A1 |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(X > 43.2) = P\!\left(Z > \dfrac{43.2 - 41.2}{3.6}\right) = P(Z > 0.5556)$ | M1 | Use of $\pm$standardisation formula once, allow continuity correction, not $\sigma^2$, $\sqrt{\sigma}$. |
| $1 - \Phi(0.5556) = 1 - 0.7108$ | M1 | Appropriate area $\Phi$, from final process, must be probability. |
| $0.289$ | A1 | AWRT |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Probability $= 1 - \textit{their}\ \textbf{(a)} = 1 - 0.2892 = 0.7108$ | B1FT | $1 - \textit{their}\ \textbf{(a)}$ or correct. |
| $0.7108 \times 365 = 259.4$; answer: $259, 260$ | B1FT | FT *their* 4SF (or better) probability, final answer must be positive integer. |
---
## Question 4(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $z = \pm 1.645$ | B1 | CAO, critical $z$ value. |
| $\dfrac{t - 41.2}{3.6} = -1.645$ | M1 | Use of $\pm$standardisation formula with $\mu$, $\sigma$ equated to a $z$-value, no continuity correction, allow $\sigma^2$, $\sqrt{\sigma}$. |
| $t = 35.3$ | A1 | |
---4 Raj wants to improve his fitness, so every day he goes for a run. The times, in minutes, of his runs have a normal distribution with mean 41.2 and standard deviation 3.6.
\begin{enumerate}[label=(\alph*)]
\item Find the probability that on a randomly chosen day Raj runs for more than 43.2 minutes.
\item Find an estimate for the number of days in a year ( 365 days) on which Raj runs for less than 43.2 minutes.
\item On 95\% of days, Raj runs for more than $t$ minutes.
Find the value of $t$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2021 Q4 [8]}}