Edexcel S3 (Statistics 3) 2013 June

1. A gym club has 400 members of which 300 are males.

Explain clearly how a stratified sample of size 60 could be taken.

2. A random sample of size \(n\) is to be taken from a population that is normally distributed with mean 40 and standard deviation 3 . Find the minimum sample size such that the probability of the sample mean being greater than 42 is less than \(5 \%\).

3. The table below shows the population and the number of council employees for different towns and villages.

1. Find, to 3 decimal places, Spearman's rank correlation coefficient between the population and the number of council employees.
2. Use your value of Spearman's rank correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a \(2.5 \%\) significance level. State your hypotheses clearly. It is suggested that a product moment correlation coefficient would be a more suitable calculation in this case. The product moment correlation coefficient for these data is 0.627 to 3 decimal places.
3. Use the value of the product moment correlation coefficient to test for evidence of a positive correlation between the population and the number of council employees. Use a \(2.5 \%\) significance level.
4. Interpret and comment on your results from part(b) and part(c).

1. John thinks that a person's eye colour is related to their hair colour. He takes a random sample of 600 people and records their eye and hair colours. The results are shown in Table 1.

\begin{table}[h] \end{table} John carries out a \(\chi ^ { 2 }\) test in order to test whether eye colour and hair colour are related. He calculates the expected frequencies shown in Table 2. \begin{table}[h] \end{table}

1. Show how the value 47 in Table 2 has been calculated.
2. Write down the number of degrees of freedom John should use in this \(\chi ^ { 2 }\) test. Given that the value of the \(\chi ^ { 2 }\) statistic is 20.6 , to 3 significant figures,
3. find the smallest value of \(\alpha\) for which the null hypothesis will be rejected at the \(\alpha \%\) level of significance.
4. Use the data from Table 1 to test at the \(5 \%\) level of significance whether or not the proportions of people in the population with black, brown, red and blonde hair are in the ratio 2:6:1:3 State your hypotheses clearly.

1. A manufacturer produces circular discs with diameter \(D \mathrm {~mm}\), such that \(D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\). A random sample of discs is taken and, using tables of the normal distribution, a \(90 \%\) confidence interval for \(\mu\) is found to be
   (118.8, 121.2)
   1. Find a 98\% confidence interval for \(\mu\).
   2. Hence write down a 98\% confidence interval for the circumference of the discs.
   Using three different random samples, three \(98 \%\) confidence intervals for \(\mu\) are to be found.
2. Calculate the probability that all the intervals will contain \(\mu\).

6. The continuous random variable \(X\) is uniformly distributed over the interval $$[ a - 1 , a + 5 ]$$ where \(a\) is a constant.
Fifty observations of \(X\) are taken, giving a sample mean of 17.2

1. Use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\).
2. Hence find a 95\% confidence interval for \(a\).

7. A farmer monitored the amount of lead in soil in a field next to a factory. He took 100 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(67 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(25 \mathrm { mg } / \mathrm { kg }\).
After the factory closed, the farmer took 150 samples of soil, randomly selected from different parts of the field, and found the mean weight of lead to be \(60 \mathrm { mg } / \mathrm { kg }\) with standard deviation \(10 \mathrm { mg } / \mathrm { kg }\).

1. Test at the \(5 \%\) level of significance whether or not the mean weight of lead in the soil decreased after the factory closed. State your hypotheses clearly.
2. Explain the significance of the Central Limit Theorem to the test in part(a).
3. State an assumption you have made to carry out this test.

8. A farmer supplies both duck eggs and chicken eggs. The weights of duck eggs, \(D\) grams, and chicken eggs, \(C\) grams, are such that $$D \sim \mathrm {~N} \left( 54,1.2 ^ { 2 } \right) \text { and } C \sim \mathrm {~N} \left( 44,0.8 ^ { 2 } \right)$$

1. Find the probability that the weights of 2 randomly selected duck eggs will differ by more than 3 g .
2. Find the probability that the weight of a randomly selected chicken egg is less than \(\frac { 4 } { 5 }\) of the weight of a randomly selected duck egg. Eggs are packed in boxes which contain either 6 randomly selected duck eggs or 6 randomly selected chicken eggs. The weight of an empty box has distribution \(\mathrm { N } \left( 28 , \sqrt { 5 } ^ { 2 } \right)\).
3. Find the probability that a full box of duck eggs weighs at least 50 g more than a full box of chicken eggs.