| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/3 (Pre-U Mathematics Paper 3) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Topic | Normal Distribution |
| Type | Standard two probabilities given |
| Difficulty | Standard +0.3 This is a standard inverse normal distribution problem requiring students to set up two equations using z-scores from given probabilities, solve simultaneously for mean and standard deviation, then apply a shift to the distribution. While it involves multiple steps and careful algebraic manipulation, the techniques are routine for A-level statistics with no novel insight required—slightly easier than average due to its straightforward structure. |
| Spec | 2.04e Normal distribution: as model N(mu, sigma^2)2.04f Find normal probabilities: Z transformation5.02c Linear coding: effects on mean and variance |
**Question 5(i)**
$1.645$ or $1.96(0)$ seen **B1**
Standardise $500$ or $495$ **M1**
$$\frac{500 - \mu}{\sigma} = 1.645$$ **A1**
$$\frac{495 - \mu}{\sigma} = -1.96(0)$$ **A1**
Solve for $\mu$ or $\sigma$ **M1** Using equations not involving probabilities, must be z values
$\mu = 497.7$, $\sigma = 1.387$ **A1** [Exact values are $\mu = \frac{51265}{103}$ and $\sigma = \frac{1000}{721}$]
**Question 5(ii)**
Use $\mu' = 500.7$ and $\sigma = 1.387$ **B1** **FT** on *their* $\mu$ and $\sigma$
Standardise $500$ using their $\mu'$ and $\sigma$ **M1** Allow with $\mu' = \mu \pm 3$
Get $0.693(2)$ **A1** From exact values 0.6977; cao5 A soft drinks company has an automated bottling machine that fills 500 ml bottles with soft drink. The contents of the bottles are measured during a check on the machine. In the check, $5 \%$ of the bottles contain more than 500 ml and $2.5 \%$ contain less than 495 ml . It is given that the amount of drink dispensed per bottle is normally distributed.\\
(i) Find the mean and standard deviation of the amount of drink dispensed per bottle, giving your answers to 4 significant figures.\\
(ii) It is subsequently found that the measurements of volume made in the checking process are all 3 ml below their true value. Using a corrected distribution, find the probability that a bottle chosen at random contains more than 500 ml of the drink.
\hfill \mbox{\textit{Pre-U Pre-U 9794/3 2018 Q5 [9]}}