Edexcel S3 (Statistics 3) 2016 June

1. The table below shows the distance travelled by car and the amount of commission earned by each of 8 salespersons in 2015

1. Find Spearman's rank correlation coefficient for these data.
2. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the distance travelled by car and the amount of commission earned.

2. A researcher investigates the results of candidates who took their driving test at one of three driving test centres. A random sample of 620 candidates gave the following results.

1. Test, at the \(5 \%\) level of significance, whether there is an association between the results of candidates' driving tests and the driving test centre. State your hypotheses and show your working clearly. You should state your expected frequencies correct to 2 decimal places. The researcher decides to conduct a further investigation into the results of candidates' driving tests.
2. State which driving test centre you would recommend for further investigation. Give a reason for your answer.

3. A company wants to survey its employees' attitudes to work. The company's workforce is located at three offices. The number of employees at each location is summarised in the table below. Each employee is located at only one office. A personnel assistant plans to survey the first 50 employees who arrive for work at the Bristol office on a Monday morning.

1. Give two reasons why this survey is likely to lead to a biased response. A personnel manager has access to the company's information system that holds details of each employee including their place of work. The manager decides to take a stratified sample of 150 employees.
2. Describe how to choose employees for this stratified sample.
3. Explain an advantage of using a stratified sample rather than a quota sample.

4. A random sample of 60 children and a random sample of 50 adults were taken and each person was given the same task to complete. The table below summarises the times taken, \(t\) seconds, to complete the task.

1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not there is evidence that the mean time taken to complete the task by children is greater than the mean time taken by adults.
   (6)
2. Explain the relevance of the Central Limit Theorem to your calculation in part (a).
3. State an assumption you have made to carry out the test in part (a).

5. Kylie used video technology to monitor the direction of flight, as a bearing, \(x\) degrees, for 450 honeybees that left her beehive during a particular morning. Kylie's results are summarised in the table below. Kylie believes that a continuous uniform distribution over the interval [0,360] is a suitable model for the direction of flight. Stating your hypotheses clearly, use a 1\% level of significance to test Kylie's belief. Show your working clearly.

6. The random variable \(W\) is defined as $$W = 3 X - 4 Y$$ where \(X \sim \mathrm {~N} \left( 21,2 ^ { 2 } \right)\) and \(Y \sim \mathrm {~N} \left( 8.5 , \sigma ^ { 2 } \right)\) and \(X\) and \(Y\) are independent.
Given that \(\mathrm { P } ( W < 44 ) = 0.9\)

1. find the value of \(\sigma\), giving your answer to 2 decimal places. The random variables \(A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) each have the same distribution as \(A\), where \(A \sim \mathrm {~N} \left( 28,5 ^ { 2 } \right)\) The random variable \(B\) is defined as $$B = 2 X + \sum _ { i = 1 } ^ { 3 } A _ { i }$$ where \(X , A _ { 1 } , A _ { 2 }\) and \(A _ { 3 }\) are independent.
2. Find \(\mathrm { P } ( B \leqslant 145 \mid B > 120 )\)

7. A random sample of 8 apples is taken from an orchard and the weight, in grams, of each apple is measured. The results are given below. $$\begin{array} { l l l l l l l l } 143 & 131 & 165 & 122 & 137 & 155 & 148 & 151 \end{array}$$

1. Calculate unbiased estimates for the mean and the variance of the weights of apples. A population has an unknown mean \(\mu\) and an unknown variance \(\sigma ^ { 2 }\)
   A random sample represented by \(X _ { 1 } , X _ { 2 } , X _ { 3 } , \ldots , X _ { 8 }\) is taken from this population.
2. Explain why \(\sum _ { i = 1 } ^ { 8 } \left( X _ { i } - \mu \right) ^ { 2 }\) is not a statistic. Given that \(\mathrm { E } \left( S ^ { 2 } \right) = \sigma ^ { 2 }\), where \(S ^ { 2 }\) is an unbiased estimator of \(\sigma ^ { 2 }\) and the statistic $$Y = \frac { 1 } { 8 } \left( \sum _ { i = 1 } ^ { 8 } X _ { i } ^ { 2 } - 8 \bar { X } ^ { 2 } \right)$$
3. find \(\mathrm { E } ( Y )\) in terms of \(\sigma ^ { 2 }\)
4. Hence find the bias, in terms of \(\sigma ^ { 2 }\), when \(Y\) is used as an estimator of \(\sigma ^ { 2 }\)
   

8. A six-sided die is labelled with the numbers \(1,2,3,4,5\) and 6 A group of 50 students want to test whether or not the die is fair for the number six.
The 50 students each roll the die 30 times and record the number of sixes they each obtain.
Given that \(\bar { X }\) denotes the mean number of sixes obtained by the 50 students, and using $$\mathrm { H } _ { 0 } : p = \frac { 1 } { 6 } \text { and } \mathrm { H } _ { 1 } : p \neq \frac { 1 } { 6 }$$ where \(p\) is the probability of rolling a 6 ,

1. use the Central Limit Theorem to find an approximate distribution for \(\bar { X }\), if \(\mathrm { H } _ { 0 }\) is true.
2. Hence find, in terms of \(\bar { X }\), the critical region for this test. Use a \(5 \%\) level of significance.