| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2005 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Probability calculation plus find unknown boundary |
| Difficulty | Moderate -0.8 This is a straightforward S1 normal distribution question requiring standard z-score calculations and inverse normal lookup. All three parts are routine textbook exercises: (a) and (b) involve converting to z-scores and using tables, while (c) is a standard inverse normal problem. No problem-solving or conceptual insight required beyond direct application of the normal distribution formula. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(M > 160) = P\left(z > \frac{160-155}{3.5}\right)\) | M1 | Standardising \(\pm(160-155)\), \(\sigma\), \(\sigma^2\), \(\sqrt{\sigma}\) |
| \(= P(z > 1.43)\) | A1 | |
| \(= 0.0764\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(150 \leq M \leq 157) = P(-1.43 \leq z \leq 0.57)\) | B1 B1 | awrt \(-1.43\), \(0.57\); \(p > 0.5\) |
| \(= 0.7157 - (1 - 0.9236)\) | M1 | |
| \(= 0.6393\) | A1 | \(0.6393 - 0.6400\) 4dp |
| Answer | Marks | Guidance |
|---|---|---|
| \(P(M \leq m) = 0.3 \Rightarrow \frac{m - 155}{3.5} = -0.5244\) | B1 | \(-0.5244\); att stand = \(z\) value; for A1 may use awrt \(-0.52\) |
| M1 A1 | ||
| \(m = 153.2\) | A1 | cao |
# Question 6:
## Part (a)
$M \sim N(155, 3.5^2)$
$P(M > 160) = P\left(z > \frac{160-155}{3.5}\right)$ | M1 | Standardising $\pm(160-155)$, $\sigma$, $\sigma^2$, $\sqrt{\sigma}$
$= P(z > 1.43)$ | A1 |
$= 0.0764$ | A1 |
## Part (b)
$P(150 \leq M \leq 157) = P(-1.43 \leq z \leq 0.57)$ | B1 B1 | awrt $-1.43$, $0.57$; $p > 0.5$
$= 0.7157 - (1 - 0.9236)$ | M1 |
$= 0.6393$ | A1 | $0.6393 - 0.6400$ 4dp
Special case: answer only B0 B0 M1 A1
## Part (c)
$P(M \leq m) = 0.3 \Rightarrow \frac{m - 155}{3.5} = -0.5244$ | B1 | $-0.5244$; att stand = $z$ value; for A1 may use awrt $-0.52$
| M1 A1 |
$m = 153.2$ | A1 | cao
---\begin{enumerate}
\item A scientist found that the time taken, $M$ minutes, to carry out an experiment can be modelled by a normal random variable with mean 155 minutes and standard deviation 3.5 minutes.
\end{enumerate}
Find\\
(a) $\mathrm { P } ( M > 160 )$.\\
(b) $\mathrm { P } ( 150 \leqslant M \leqslant 157 )$.\\
(c) the value of $m$, to 1 decimal place, such that $\mathrm { P } ( M \leqslant m ) = 0.30$.\\
\hfill \mbox{\textit{Edexcel S1 2005 Q6 [11]}}