Edexcel S3 (Statistics 3) 2004 June

1. There are 64 girls and 56 boys in a school.

Explain briefly how you could take a random sample of 15 pupils using

1. a simple random sample,
2. a stratified sample.

2. A random sample of 8 students sat examinations in Geography and Statistics. The product moment correlation coefficient between their results was 0.572 and the Spearman rank correlation coefficient was 0.655 .

1. Test both of these values for positive correlation. Use a \(5 \%\) level of significance.
2. Comment on your results.

3. It is known from past evidence that the weight of coffee dispensed into jars by machine \(A\) is normally distributed with mean \(\mu _ { \mathrm { A } }\) and standard deviation 2.5 g . Machine \(B\) is known to dispense the same nominal weight of coffee into jars with mean \(\mu _ { B }\) and standard deviation 2.3 g . A random sample of 10 jars filled by machine \(A\) contained a mean weight of 249 g of coffee. A random sample of 15 jars filled by machine \(B\) contained a mean weight of 251 g .

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the population mean weight dispensed by machine B is greater than that of machine A .
2. Write down an assumption needed to carry out this test.

4. Kylie regularly travels from home to visit a friend. On 10 randomly selected occasions the journey time \(x\) minutes was recorded. The results are summarised as follows. $$\Sigma x = 753 , \quad \Sigma x ^ { 2 } = 57455 .$$

1. Calculate unbiased estimates of the mean and the variance of the population of journey times. After many journeys, a random sample of 100 journeys gave a mean of 74.8 minutes and a variance of 84.6 minutes \({ } ^ { 2 }\).
2. Calculate a 95\% confidence interval for the mean of the population of journey times.
3. Write down two assumptions you made in part (b).

5. A random sample of 500 adults completed a questionnaire on how often they took part in some form of exercise. They gave a response of 'never', 'sometimes' or 'regularly'. Of those asked, \(52 \%\) were females of whom \(10 \%\) never exercised and \(35 \%\) exercised regularly. Of the males, \(12.5 \%\) never exercised and \(55 \%\) sometimes exercised. Test, at the \(5 \%\) level of significance, whether or not there is any association between gender and the amount of exercise. State your hypotheses clearly.

6 Three six-sided dice, which were assumed to be fair, were rolled 250 times. On each occasion the number \(X\) of sixes was recorded. The results were as follows.

1. Write down a suitable model for \(X\).
2. Test, at the \(1 \%\) level of significance, the suitability of your model for these data.
3. Explain how the test would have been modified if it had not been assumed that the dice were fair.

7. The random variable \(D\) is defined as $$D = A - 3 B + 4 C$$ where \(A \sim \mathrm {~N} \left( 5,2 ^ { 2 } \right) , B \sim \mathrm {~N} \left( 7,3 ^ { 2 } \right)\) and \(C \sim \mathrm {~N} \left( 9,4 ^ { 2 } \right)\), and \(A , B\) and \(C\) are independent.

1. Find \(\mathrm { P } ( \mathrm { D } < 44 )\). The random variables \(B _ { 1 } , B _ { 2 }\) and \(B _ { 3 }\) are independent and each has the same distribution as \(B\). The random variable \(X\) is defined as $$X = A - \sum _ { i = 1 } ^ { 3 } B _ { i } + 4 C .$$
2. Find \(\mathrm { P } ( X > 0 )\). \section*{END}