Edexcel S3 (Statistics 3) 2007 June

1. During a village show, two judges, \(P\) and \(Q\), had to award a mark out of 30 to some flower displays. The marks they awarded to a random sample of 8 displays were as follows:

1. Calculate Spearman's rank correlation coefficient for the marks awarded by the two judges. After the show, one competitor complained about the judges. She claimed that there was no positive correlation between their marks.
2. Stating your hypotheses clearly, test whether or not this sample provides support for the competitor's claim. Use a \(5 \%\) level of significance.

1. The Director of Studies at a large college believed that students' grades in Mathematics were independent of their grades in English. She examined the results of a random group of candidates who had studied both subjects and she recorded the number of candidates in each of the 6 categories shown.

1. Stating your hypotheses clearly, test the Director's belief using a \(10 \%\) level of significance. You must show each step of your working. The Head of English suggested that the Director was losing accuracy by combining the English grades C to U in one row. He suggested that the Director should split the English grades into two rows, grades C or D and grades E or U as for Mathematics.
2. State why this might lead to problems in performing the test.

1. The time, in minutes, it takes Robert to complete the puzzle in his morning newspaper each day is normally distributed with mean 18 and standard deviation 3. After taking a holiday, Robert records the times taken to complete a random sample of 15 puzzles and he finds that the mean time is 16.5 minutes. You may assume that the holiday has not changed the standard deviation of times taken to complete the puzzle.

Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there has been a reduction in the mean time Robert takes to complete the puzzle.

4. A quality control manager regularly samples 20 items from a production line and records the number of defective items \(x\). The results of 100 such samples are given in table 1 below. \begin{table}[h] \end{table}

1. Estimate the proportion of defective items from the production line. The manager claimed that the number of defective items in a sample of 20 can be modelled by a binomial distribution. He used the answer in part (a) to calculate the expected frequencies given in Table 2. \begin{table}[h]

   \(x\)	0	1	2	3	4	5	6	7 or more
   Expected frequency	12.2	27.0	\(r\)	19.0	\(s\)	3.2	0.9	0.2

   \captionsetup{labelformat=empty} \caption{Table 2}
   \end{table}
2. Find the value of \(r\) and the value of \(s\) giving your answers to 1 decimal place.
3. Stating your hypotheses clearly, use a \(5 \%\) level of significance to test the manager's claim.
4. Explain what the analysis in part (c) tells the manager about the occurrence of defective items from this production line.

1. In a trial of \(\operatorname { diet } A\) a random sample of 80 participants were asked to record their weight loss, \(x \mathrm {~kg}\), after their first week of using the diet. The results are summarised by

$$\sum x = 361.6 \text { and } \sum x ^ { 2 } = 1753.95$$

1. Find unbiased estimates for the mean and variance of weight lost after the first week of using diet \(A\). The designers of diet \(A\) believe it can achieve a greater mean weight loss after the first week than a standard diet \(B\). A random sample of 60 people used diet \(B\). After the first week they had achieved a mean weight loss of 4.06 kg , with an unbiased estimate of variance of weight loss of \(2.50 \mathrm {~kg} ^ { 2 }\).
2. Test, at the \(5 \%\) level of significance, whether or not the mean weight loss after the first week using \(\operatorname { diet } A\) is greater than that using diet \(B\). State your hypotheses clearly.
3. Explain the significance of the central limit theorem to the test in part (b).
4. State an assumption you have made in carrying out the test in part (b).

1. A random sample of the daily sales (in £s) of a small company is taken and, using tables of the normal distribution, a 99\% confidence interval for the mean daily sales is found to be
   (123.5, 154.7)

Find a \(95 \%\) confidence interval for the mean daily sales of the company.
(6)

7. A set of scaffolding poles come in two sizes, long and short. The length \(L\) of a long pole has the normal distribution \(\mathrm { N } \left( 19.7,0.5 ^ { 2 } \right)\). The length \(S\) of a short pole has the normal distribution \(\mathrm { N } \left( 4.9,0.2 ^ { 2 } \right)\). The random variables \(L\) and \(S\) are independent. A long pole and a short pole are selected at random.

1. Find the probability that the length of the long pole is more than 4 times the length of the short pole. Four short poles are selected at random and placed end to end in a row. The random variable \(T\) represents the length of the row.
2. Find the distribution of \(T\).
3. Find \(\mathrm { P } ( | L - T | < 0.1 )\).