| Exam Board | Edexcel |
|---|---|
| Module | S1 (Statistics 1) |
| Year | 2003 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Normal Distribution |
| Type | Direct expected frequency calculation |
| Difficulty | Moderate -0.8 This is a straightforward application of normal distribution with standard procedures: (a) requires a basic z-score calculation and multiplication by sample size, while (b) involves inverse normal lookup. All steps are routine S1 techniques with no problem-solving insight needed, making it easier than average but not trivial due to the inverse normal component. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation |
| Answer | Marks | Guidance |
|---|---|---|
| Part (a)(i) | ||
| Let \(X\) represent amount of sauce in a jar. \(\therefore X \sim N(505, 10^2)\) | ||
| \(\therefore P[X < 500] = P\left(Z < \frac{500 - 505}{10}\right)\) | M1 | Standardising with 505, 10 |
| \(= P(Z < -0.5)\) | A1 | \(-0.5\) |
| \(= 1 - 0.6915\) | ||
| \(= 0.3085\) | A1 | (5 marks) |
| Part (a)(ii) | ||
| Expected number \(= 30 \times 0.3085\) | M1 | \(30 \times (f)\) |
| \(= 9.225\) | A1 | \(9.23\) |
| Part (b) | ||
| \(P[X < 500] = 0.01\) | B1 | |
| \(\therefore \frac{500 - \mu}{10} = -2.3263\) | M1 | Standardising |
| B1 | \(-2.3263\) | |
| \(\therefore \mu = 523.263\) | ||
| \(523\) | A1 | (4 marks) |
**Part (a)(i)** | |
Let $X$ represent amount of sauce in a jar. $\therefore X \sim N(505, 10^2)$ | |
$\therefore P[X < 500] = P\left(Z < \frac{500 - 505}{10}\right)$ | M1 | Standardising with 505, 10 |
$= P(Z < -0.5)$ | A1 | $-0.5$ |
$= 1 - 0.6915$ | |
$= 0.3085$ | A1 | (5 marks)
**Part (a)(ii)** | |
Expected number $= 30 \times 0.3085$ | M1 | $30 \times (f)$ |
$= 9.225$ | A1 | $9.23$ |
**Part (b)** | |
$P[X < 500] = 0.01$ | B1 |
$\therefore \frac{500 - \mu}{10} = -2.3263$ | M1 | Standardising |
| | B1 | $-2.3263$ |
$\therefore \mu = 523.263$ | |
$523$ | A1 | (4 marks)
---3. Cooking sauces are sold in jars containing a stated weight of 500 g of sauce The jars are filled by a machine. The actual weight of sauce in each jar is normally distributed with mean 505 g and standard deviation 10 g .
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the probability of a jar containing less than the stated weight.
\item In a box of 30 jars, find the expected number of jars containing less than the stated weight.
The mean weight of sauce is changed so that $1 \%$ of the jars contain less than the stated weight. The standard deviation stays the same.
\end{enumerate}\item Find the new mean weight of sauce.
\end{enumerate}
\hfill \mbox{\textit{Edexcel S1 2003 Q3 [9]}}