OCR MEI S1 (Statistics 1) 2010 June

1 A business analyst collects data about the distribution of hourly wages, in \(\pounds\), of shop-floor workers at a factory. These data are illustrated in the box and whisker plot.
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1. Name the type of skewness of the distribution.
2. Find the interquartile range and hence show that there are no outliers at the lower end of the distribution, but there is at least one outlier at the upper end.
3. Suggest possible reasons why this may be the case.

2 The probability distribution of the random variable \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k r ( 5 - r ) \text { for } r = 1,2,3,4 .$$

1. Show that \(k = 0.05\).
2. Find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).

3 The lifetimes in hours of 90 components are summarised in the table.

1. Draw a histogram to illustrate these data.
2. In which class interval does the median lie? Justify your answer.

4 Each packet of Cruncho cereal contains one free fridge magnet. There are five different types of fridge magnet to collect. They are distributed, with equal probability, randomly and independently in the packets. Keith is about to start collecting these fridge magnets.

1. Find the probability that the first 2 packets that Keith buys contain the same type of fridge magnet.
2. Find the probability that Keith collects all five types of fridge magnet by buying just 5 packets.
3. Hence find the probability that Keith has to buy more than 5 packets to acquire a complete set.

5 A retail analyst records the numbers of loaves of bread of a particular type bought by a sample of shoppers in a supermarket.

1. Calculate the mean and standard deviation of the numbers of loaves bought per person.
2. Each loaf costs \(\pounds 1.04\). Calculate the mean and standard deviation of the amount spent on loaves per person.

6 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.

1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
   (B) Find the probability that there are more than 2 faulty tiles in the sample.
   (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
2. (A) Write down suitable null and alternative hypotheses for the test.
   (B) Explain why the alternative hypothesis has the form that it does.
3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.

7 One train leaves a station each hour. The train is either on time or late. If the train is on time, the probability that the next train is on time is 0.95 . If the train is late, the probability that the next train is on time is 0.6 . On a particular day, the 0900 train is on time.

1. Illustrate the possible outcomes for the 1000,1100 and 1200 trains on a probability tree diagram.
2. Find the probability that
   (A) all three of these trains are on time,
   (B) just one of these three trains is on time,
   (C) the 1200 train is on time.
3. Given that the 1200 train is on time, find the probability that the 1000 train is also on time. 3
4. Write any calculations on page 5.
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