OCR MEI S3 (Statistics 3) 2006 January

1 A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 3 } t } & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay.
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay.
3. Find the probability that a delay lasts longer than the mean delay. You are given that the variance of \(T\) is 9 .
4. Let \(\bar { T }\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\bar { T }\).
5. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling?

3 A production line has two machines, A and B , for delivering liquid soap into bottles. Each machine is set to deliver a nominal amount of 250 ml , but it is not expected that they will work to a high level of accuracy. In particular, it is known that the ambient temperature affects the rate of flow of the liquid and leads to variation in the amounts delivered. The operators think that machine B tends to deliver a somewhat greater amount than machine A , no matter what the ambient temperature. This is being investigated by an experiment. A random sample of 10 results from the experiment is shown below. Each column of data is for a different ambient temperature.

1. Use an appropriate \(t\) test to examine, at the \(5 \%\) level of significance, whether the mean amount delivered by machine B may be taken as being greater than that delivered by machine A , stating carefully your null and alternative hypotheses and the required distributional assumption.
2. Using the data for machine A in the table above, provide a two-sided \(95 \%\) confidence interval for the mean amount delivered by this machine, stating the required distributional assumption. Explain whether you would conclude that the machine appears to be working correctly in terms of the nominal amount as set.

4 Quality control inspectors in a factory are investigating the lengths of glass tubes that will be used to make laboratory equipment.

1. Data on the observed lengths of a random sample of 200 glass tubes from one batch are available in the form of a frequency distribution as follows.

   Length \(x ( \mathrm {~mm} )\)	Observed frequency
   \(x \leqslant 298\)	1
   \(298 < x \leqslant 300\)	30
   \(300 < x \leqslant 301\)	62
   \(301 < x \leqslant 302\)	70
   \(302 < x \leqslant 304\)	34
   \(x > 304\)	3

   The sample mean and standard deviation are 301.08 and 1.2655 respectively.
   The corresponding expected frequencies for the Normal distribution with parameters estimated by the sample statistics are

   Length \(x ( \mathrm {~mm} )\)	Expected frequency
   \(x \leqslant 298\)	1.49
   \(298 < x \leqslant 300\)	37.85
   \(300 < x \leqslant 301\)	55.62
   \(301 < x \leqslant 302\)	58.32
   \(302 < x \leqslant 304\)	44.62
   \(x > 304\)	2.10

   Examine the goodness of fit of a Normal distribution, using a 5\% significance level.
2. It is thought that the lengths of tubes in another batch have an underlying distribution similar to that for the batch in part (i) but possibly with different location and dispersion parameters. A random sample of 10 tubes from this batch gives the following lengths (in mm ). $$\begin{array} { l l l l l l l l l l } 301.3 & 301.4 & 299.6 & 302.2 & 300.3 & 303.2 & 302.6 & 301.8 & 300.9 & 300.8 \end{array}$$ (A) Discuss briefly whether it would be appropriate to use a \(t\) test to examine a hypothesis about the population mean length for this batch.
   (B) Use a Wilcoxon test to examine at the \(10 \%\) significance level whether the population median length for this batch is 301 mm .