OCR MEI S3 (Statistics 3) 2011 June

1 Gerry runs 5000 -metre races for his local athletics club. His coach has been monitoring his practice times for several months and he believes that they can be modelled using a Normal distribution with mean 15.3 minutes. The coach suggests that Gerry should try running with a pacemaker in order to see if this can improve his times. Subsequently a random sample of 10 of Gerry's times with the pacemaker is collected to see if any reduction has been achieved. The sample of times (in minutes) is as follows. $$\begin{array} { l l l l l l l l l l } 14.86 & 15.00 & 15.62 & 14.44 & 15.27 & 15.64 & 14.58 & 14.30 & 15.08 & 15.08 \end{array}$$

1. Why might a \(t\) test be used for these data?
2. Using a \(5 \%\) significance level, carry out the test to see whether, on average, Gerry's times have been reduced.
3. What is meant by 'a \(5 \%\) significance level'? What would be the consequence of decreasing the significance level?
4. Find a \(95 \%\) confidence interval for the true mean of Gerry's times using a pacemaker.

2 Scientists researching into the chemical composition of dust in space collect specimens using a specially designed spacecraft. The craft collects the particles of dust in trays that are made up of a large array of cells containing aerogel. The aerogel traps the particles that penetrate into the cells.

1. For a random sample of 100 cells, the number of particles of dust in each cell was counted, giving the following results.

   Number of particles	0	1	2	3	4	5	6	7	8	9	\(10 +\)
   Frequency	4	7	10	20	17	15	10	9	5	3	0

   It is thought that the number of particles collected in each cell can be modelled using the distribution Poisson(4.2) since 4.2 is the sample mean for these data. Some of the calculations for a \(\chi ^ { 2 }\) test are shown below. The cells for 8,9 and \(10 +\) particles have been combined.

   Number of particles

   Observed frequency

   Expected frequency

   Contribution to \(X ^ { 2 }\)

   5	6	7	\(8 +\)
   15	10	9	8
   16.33	11.44	6.86	6.39
   0.1083	0.1813	0.6676	0.4056

   Complete the calculations and carry out the test using a \(10 \%\) significance level to see whether the number of particles per cell may be modelled in this way.
2. The diameters of the dust particles are believed to be distributed symmetrically about a median of 15 micrometres \(( \mu \mathrm { m } )\). For a random sample of 20 particles, the sum of the signed ranks of the diameters of the particles smaller than \(15 \mu \mathrm {~m} \left( W _ { - } \right)\)is found to be 53 . Test at the \(5 \%\) level of significance whether the median diameter appears to be more than \(15 \mu \mathrm {~m}\).

3 The time, in hours, until an electronic component fails is represented by the random variable \(X\). In this question two models for \(X\) are proposed.

1. In one model, \(X\) has cumulative distribution function $$\mathrm { G } ( x ) = \begin{cases} 0 & x \leqslant 0
   1 - \left( 1 + \frac { x } { 200 } \right) ^ { - 2 } & x > 0 \end{cases}$$ (A) Sketch \(\mathrm { G } ( x )\).
   (B) Find the interquartile range for this model. Hence show that a lifetime of more than 454 hours (to the nearest hour) would be classed as an outlier.
2. In the alternative model, \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 200 } \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0
   0 & \text { elsewhere. } \end{cases}$$ (A) For this model show that the cumulative distribution function of \(X\) is $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0
   1 - \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \end{cases}$$ (B) Show that \(\mathrm { P } ( X > 50 ) = \mathrm { e } ^ { - 0.25 }\).
   (C) It is observed that a particular component is still working after 400 hours. Find the conditional probability that it will still be working after a further 50 hours (i.e. after a total of 450 hours) given that it is still working after 400 hours.

4 The weights of Avonley Blue cheeses made by a small producer are found to be Normally distributed with mean 10 kg and standard deviation 0.4 kg .

1. Find the probability that a randomly chosen cheese weighs less than 9.5 kg . One particular shop orders four Avonley Blue cheeses each week from the producer. From experience, the shopkeeper knows that the weekly demand from customers for Avonley Blue cheese is Normally distributed with mean 35 kg and standard deviation 3.5 kg . In the interests of food hygiene, no cheese is kept by the shopkeeper from one week to the next.
2. Find the probability that, in a randomly chosen week, demand from customers for Avonley Blue will exceed the supply. Following a campaign to promote Avonley Blue cheese, the shopkeeper finds that the weekly demand for it has increased by \(30 \%\) (i.e. the mean and standard deviation are both increased by \(30 \%\) ). Therefore the shopkeeper increases his weekly order by one cheese.
3. Find the probability that, in a randomly chosen week, demand will now exceed supply.
4. Following complaints, the cheese producer decides to check the mean weight of the Avonley Blue cheeses. For a random sample of 12 cheeses, she finds that the mean weight is 9.73 kg . Assuming that the population standard deviation of the weights is still 0.4 kg , find a \(95 \%\) confidence interval for the true mean weight of the cheeses and comment on the result. Explain what is meant by a 95\% confidence interval. RECOGNISING ACHIEVEMENT