| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Topic | Normal Distribution |
| Type | Mixed calculations with boundaries |
| Difficulty | Standard +0.3 This is a standard normal distribution application requiring inverse normal calculations and hypothesis testing. Part (i) involves setting up two simultaneous equations using z-scores (routine but multi-step), part (ii) is a direct percentile calculation, and part (iii) requires comparing observed vs expected proportions. While it requires careful work with tables/calculator, the techniques are well-practiced and the question follows a familiar template for A-level statistics problems. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use \(z\) values of −1.406 AND 1.036 or 1.037 | B1 | |
| Use their \(z_1 = \frac{15-\mu}{\sigma}\) and their \(z_2 = \frac{12-\mu}{\sigma}\) | M1 | |
| \(1.036 = \frac{15-\mu}{\sigma}\) and \(-1.406 = \frac{12-\mu}{\sigma}\) | A1 | |
| Solving two simultaneous eqns in \(\mu\) and \(\sigma\) | DM1 | |
| \(\sigma = 1.23\) \(\mu = 13.7\) | A1 A1 | [6] |
| (ii) \(z = -1.645\) or 1.645 found | B1 | |
| their \(z = \frac{x-13.73}{1.229}\) | M1 | |
| 11.7 | A1 | [3] |
| (iii) \(z = \frac{16-\text{their}\mu}{\text{their}\sigma}\) | M1 | |
| Compare their value with an equivalent in the random sample | DM1 | |
| State inconsistent | A1 | [3] |
(i) Use $z$ values of −1.406 AND 1.036 or 1.037 | B1 |
Use their $z_1 = \frac{15-\mu}{\sigma}$ and their $z_2 = \frac{12-\mu}{\sigma}$ | M1 |
$1.036 = \frac{15-\mu}{\sigma}$ and $-1.406 = \frac{12-\mu}{\sigma}$ | A1 |
Solving two simultaneous eqns in $\mu$ and $\sigma$ | DM1 |
$\sigma = 1.23$ $\mu = 13.7$ | A1 A1 | [6]
(ii) $z = -1.645$ or 1.645 found | B1 |
their $z = \frac{x-13.73}{1.229}$ | M1 |
11.7 | A1 | [3]
(iii) $z = \frac{16-\text{their}\mu}{\text{their}\sigma}$ | M1 |
Compare their value with an equivalent in the random sample | DM1 |
State inconsistent | A1 | [3]A manufacturer produces components designed with length $L$ mm such that $12 < L < 15$. The Quality Control department finds that 15% of the components sampled are longer than 15 mm while 8% are shorter than 12 mm. Assume that $L$ is normally distributed with mean $\mu$ and standard deviation $\sigma$.
\begin{enumerate}[label=(\roman*)]
\item Calculate $\mu$ and $\sigma$. [6]
\item The shortest 5% of components are rejected. Find the minimum length which a component may have before it is rejected. [3]
\item It was found in a random sample that 10% of components were longer than 16 mm. Determine whether this finding is consistent with the assumption that $L$ is normally distributed with the $\mu$ and $\sigma$ found in part (i). [3]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2010 Q15 [12]}}