| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Linear relationship μ = kσ |
| Difficulty | Moderate -0.8 This is a straightforward normal distribution question requiring only standard techniques: z-score calculation for part (a)(i), inverse normal lookup for part (a)(ii), and standardization with algebraic manipulation for part (b). All are routine S1-level exercises with no problem-solving insight required, making it easier than average A-level maths questions. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.03a Continuous random variables: pdf and cdf |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) \[P(x < 8) = P\left(z < \frac{8-7.15}{0.88}\right) = \Phi(0.9659) = 0.833\] | M1, A1 2 | Standardising \(\pm\), no cc no sq rt no sq or Correct answer |
| (ii) \[z = 0.674 \quad \frac{q - 7.15}{0.88} = 0.674 \quad q = 7.74\] | B1, M1, A1 3 | Accept \(\pm 0.674\) or \(0.675\) only or Standardised eqn \(= \pm\) their z-value, allow sq or sq rt if already penalised in (i) or Correct answer |
| (b) \[P(Y > 4\mu) = P\left(z > \frac{4\mu - \mu}{(3\mu/2)}\right) = P(z > 2) = 1 - 0.9772 = 0.0228\] | M1, A1, A1 3 | Standardising no sq rt, no cc, no sq, one variable or \(z = \pm 2\) seen or correct ans SR B1 if made-up values used and 0.0228 obtained |
**(a)(i)** $$P(x < 8) = P\left(z < \frac{8-7.15}{0.88}\right) = \Phi(0.9659) = 0.833$$ | M1, A1 2 | Standardising $\pm$, no cc no sq rt no sq or Correct answer
**(ii)** $$z = 0.674 \quad \frac{q - 7.15}{0.88} = 0.674 \quad q = 7.74$$ | B1, M1, A1 3 | Accept $\pm 0.674$ or $0.675$ only or Standardised eqn $= \pm$ their z-value, allow sq or sq rt if already penalised in (i) or Correct answer
**(b)** $$P(Y > 4\mu) = P\left(z > \frac{4\mu - \mu}{(3\mu/2)}\right) = P(z > 2) = 1 - 0.9772 = 0.0228$$ | M1, A1, A1 3 | Standardising no sq rt, no cc, no sq, one variable or $z = \pm 2$ seen or correct ans SR B1 if made-up values used and 0.0228 obtained
---\begin{enumerate}[label=(\alph*)]
\item The time, $X$ hours, for which people sleep in one night has a normal distribution with mean 7.15 hours and standard deviation 0.88 hours.
\begin{enumerate}[label=(\roman*)]
\item Find the probability that a randomly chosen person sleeps for less than 8 hours in a night. [2]
\item Find the value of $q$ such that P$(X < q) = 0.75$. [3]
\end{enumerate}
\item The random variable $Y$ has the distribution N$(\mu, \sigma^2)$, where $2\sigma = 3\mu$ and $\mu \neq 0$. Find P$(Y > 4\mu)$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2014 Q5 [8]}}