Edexcel S3 (Statistics 3) 2005 June

1. (a) State two reasons why stratified sampling might be chosen as a method of sampling when carrying out a statistical survey.
   (b) State one advantage and one disadvantage of quota sampling.
2. A sample of size 5 is taken from a population that is normally distributed with mean 10 and standard deviation 3 . Find the probability that the sample mean lies between 7 and 10 .
   (Total 6 marks)
3. A researcher carried out a survey of three treatments for a fruit tree disease. The contingency table below shows the results of a survey of a random sample of 60 diseased trees.

Test, at the \(5 \%\) level of significance, whether or not there is any association between the treatment of the trees and their survival. State your hypotheses and conclusion clearly.
(Total 11 marks)

4. Over a period of time, researchers took 10 blood samples from one patient with a blood disease. For each sample, they measured the levels of serum magnesium, \(s \mathrm { mg } / \mathrm { dl }\), in the blood and the corresponding level of the disease protein, \(d \mathrm { mg } / \mathrm { dl }\). The results are shown in the table. $$\text { [Use } \sum s ^ { 2 } = 141.51 , \sum d ^ { 2 } = 1081.74 \text { and } \sum s d = 386.32 \text { ] }$$

1. Draw a scatter diagram to represent these data.
2. State what is measured by the product moment correlation coefficient.
3. Calculate \(S _ { x x } , S _ { d d }\) and \(S _ { s d }\).
4. Calculate the value of the product moment correlation coefficient \(r\) between \(s\) and \(d\).
5. Stating your hypotheses clearly, test, at the \(1 \%\) significance level, whether or not the correlation coefficient is greater than zero.
6. With reference to your scatter diagram, comment on your result in part (e).

5. The number of times per day a computer fails and has to be restarted is recorded for 200 days. The results are summarised in the table. Test whether or not a Poisson model is suitable to represent the number of restarts per day. Use a \(5 \%\) level of significance and state your hypothesis clearly.
(Total 12 marks)

6. A computer company repairs large numbers of PCs and wants to estimate the mean time to repair a particular fault. Five repairs are chosen at random from the company's records and the times taken, in seconds, are $$\begin{array} { l l l l l } 205 & 310 & 405 & 195 & 320 \end{array} .$$

1. Calculate unbiased estimates of the mean and the variance of the population of repair times from which this sample has been taken. It is known from previous results that the standard deviation of the repair time for this fault is 100 seconds. The company manager wants to ensure that there is a probability of at least 0.95 that the estimate of the population mean lies within 20 seconds of its true value.
2. Find the minimum sample size required.
   (Total 10 marks)

7. A manufacturer produces two flavours of soft drink, cola and lemonade. The weights, \(C\) and \(L\), in grams, of randomly selected cola and lemonade cans are such that \(C \sim \mathrm {~N} ( 350,8 )\) and \(L \sim \mathrm {~N} ( 345,17 )\).

1. Find the probability that the weights of two randomly selected cans of cola will differ by more than 6 g . One can of each flavour is selected at random.
2. Find the probability that the can of cola weighs more than the can of lemonade. Cans are delivered to shops in boxes of 24 cans. The weights of empty boxes are normally distributed with mean 100 g and standard deviation 2 g .
3. Find the probability that a full box of cola cans weighs between 8.51 kg and 8.52 kg .
4. State an assumption you made in your calculation in part (c).