OCR MEI S3 (Statistics 3) 2013 June

1 In the past, the times for workers in a factory to complete a particular task had a known median of 7.4 minutes. Following a review, managers at the factory wish to know if the median time to complete the task has been reduced.

1. A random sample of 12 times, in minutes, gives the following results. $$\begin{array} { l l l l l l l l l l l l } 6.90 & 7.23 & 6.54 & 7.62 & 7.04 & 7.33 & 6.74 & 6.45 & 7.81 & 7.71 & 7.50 & 6.32 \end{array}$$ Carry out an appropriate test using a \(5 \%\) level of significance.
2. Some time later, a much larger random sample of times gives the following results. $$n = 80 \quad \sum x = 555.20 \quad \sum x ^ { 2 } = 3863.9031$$ Find a \(95 \%\) confidence interval for the true mean time for the task. Justify your choice of which distribution to use.
3. Describe briefly one advantage and one disadvantage of having a \(99 \%\) confidence interval instead of a \(95 \%\) confidence interval.

2 A company supplying cattle feed to dairy farmers claims that its new brand of feed will increase average milk yields by 10 litres per cow per week. A farmer thinks the increase will be less than this and decides to carry out a statistical investigation using a paired \(t\) test. A random sample of 10 dairy cows are given the new feed and then their milk yields are compared with their yields when on the old feed. The yields, in litres per week, for the 10 cows are as follows.

1. Why is it sensible to use a paired test?
2. State the condition necessary for a paired \(t\) test.
3. Assuming the condition stated in part (ii) is met, carry out the test, using a significance level of \(5 \%\), to see whether it appears that the company's claim is justified.
4. Find a 95\% confidence interval for the mean increase in the milk yield using the new feed.

3 The random variable \(X\) has the following probability density function, \(\mathrm { f } ( x )\). $$f ( x ) = \begin{cases} k x ( x - 5 ) ^ { 2 } & 0 \leqslant x < 5
0 & \text { elsewhere } \end{cases}$$

1. Sketch \(\mathrm { f } ( x )\).
2. Find, in terms of \(k\), the cumulative distribution function, \(\mathrm { F } ( x )\).
3. Hence show that \(k = \frac { 12 } { 625 }\). The random variable \(X\) is proposed as a model for the amount of time, in minutes, lost due to stoppages during a football match. The times lost in a random sample of 60 matches are summarised in the table. The table also shows some of the corresponding expected frequencies given by the model.

   Time (minutes)	\(0 \leqslant x < 1\)	\(1 \leqslant x < 2\)	\(2 \leqslant x < 3\)	\(3 \leqslant x < 4\)	\(4 \leqslant x < 5\)
   Observed frequency	5	15	23	11	6
   Expected frequency			17.76	9.12	1.632

4. Find the remaining expected frequencies.
5. Carry out a goodness of fit test, using a significance level of \(2.5 \%\), to see if the model might be suitable in this context.

4 A company that makes meat pies includes a "small" size in its product range. These pies consist of a pastry case and meat filling, the weights of which are independent of each other. The weight of the pastry case, \(C\), is Normally distributed with mean 96 g and variance \(21 \mathrm {~g} ^ { 2 }\). The weight of the meat filling, \(M\), is Normally distributed with mean 57 g and variance \(14 \mathrm {~g} ^ { 2 }\).

1. Find the probability that, in a randomly chosen pie, the weight of the pastry case is between 90 and 100 g .
2. The wrappers on the pies state that the weight is 145 g . Find the proportion of pies that are underweight.
3. The pies are sold in packs of 4 . Find the value of \(w\) such that, in \(95 \%\) of packs, the total weight of the 4 pies in a randomly chosen pack exceeds \(w \mathrm {~g}\).
4. It is required that the weight of the meat filling in a pie should be at least \(35 \%\) of the total weight. Show that this means that \(0.65 M - 0.35 C \geqslant 0\). Hence find the probability that, in a randomly chosen pie, this requirement is met.