| Exam Board | CAIE |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Linear relationship μ = kσ |
| Difficulty | Standard +0.3 Part (a) requires standardizing with the constraint μ = 1.5σ to find P(X > 0), which is straightforward z-score calculation. Part (b) involves reverse lookup from a probability to find the standard deviation using inverse normal tables. Both are standard S1 techniques with minimal problem-solving required, slightly easier than average due to direct application of methods. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(P(x > 0) = P\!\left(z > \pm\dfrac{0 - \mu}{\sigma}\right) = P\!\left(z > \dfrac{-\mu}{\mu/1.5}\right)\) or \(P\!\left(z > \dfrac{-1.5\sigma}{\sigma}\right)\) | M1 | \(\pm\)Standardising, in terms of \(\mu\) and/or \(\sigma\) with \(0 - \ldots\) in numerator, no continuity correction, no \(\sqrt{}\) |
| \(= P(z > -1.5)\) | A1 | Obtaining \(z\) value of \(\pm 1.5\) by eliminating \(\mu\) and \(\sigma\), SOI |
| \(= 0.933\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(z = -1.151\) | B1 | \(\pm z\) value rounding to 1.1 or 1.2 |
| \(-1.151 = \dfrac{70 - 120}{s}\) | M1 | \(\pm\)Standardising (using 70) equated to a \(z\)-value, no cc, no squaring, no \(\sqrt{}\) |
| \(\sigma = 43.4\) or \(43.5\) | A1 | |
| Total: 3 |
# Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $P(x > 0) = P\!\left(z > \pm\dfrac{0 - \mu}{\sigma}\right) = P\!\left(z > \dfrac{-\mu}{\mu/1.5}\right)$ or $P\!\left(z > \dfrac{-1.5\sigma}{\sigma}\right)$ | M1 | $\pm$Standardising, in terms of $\mu$ and/or $\sigma$ with $0 - \ldots$ in numerator, no continuity correction, no $\sqrt{}$ |
| $= P(z > -1.5)$ | A1 | Obtaining $z$ value of $\pm 1.5$ by eliminating $\mu$ and $\sigma$, SOI |
| $= 0.933$ | A1 | |
| **Total: 3** | | |
# Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $z = -1.151$ | B1 | $\pm z$ value rounding to 1.1 or 1.2 |
| $-1.151 = \dfrac{70 - 120}{s}$ | M1 | $\pm$Standardising (using 70) equated to a $z$-value, no cc, no squaring, no $\sqrt{}$ |
| $\sigma = 43.4$ or $43.5$ | A1 | |
| **Total: 3** | | |4
\begin{enumerate}[label=(\alph*)]
\item The random variable $X$ has the distribution $\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)$, where $\mu = 1.5 \sigma$. A random value of $X$ is chosen. Find the probability that this value of $X$ is greater than 0 .
\item The life of a particular type of torch battery is normally distributed with mean 120 hours and standard deviation $s$ hours. It is known that $87.5 \%$ of these batteries last longer than 70 hours. Find the value of $s$.
\end{enumerate}
\hfill \mbox{\textit{CAIE S1 2017 Q4 [6]}}