Edexcel S3 (Statistics 3) 2017 June

1. A company director decides to survey staff about changes to the company calendar. The company has staff in 4 different job roles

72 managers, 108 drivers, 180 administrators and 360 warehouse staff.
The director decides to take a stratified sample.

1. Write down one advantage of using a stratified sample rather than a simple random sample for this survey.
2. Find the number of staff in each job role that will be included in a stratified sample of 40 staff.
3. Describe how to choose managers for the stratified sample.

2. \begin{figure}[h] \end{figure} The pointer shown in Figure 1 is spun so that it comes to rest between 0 and 360 degrees.
Linda claims that it is equally likely to come to rest at any point between 0 and 360 degrees. She spins the pointer 100 times and her results are summarised in the table below. She calculates expected frequencies for some of the possible outcomes and these are also given in the table below.

1. Find the values of the missing expected frequencies \(a , b\) and \(c\).
2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not Linda's claim is supported by these data.

1. A junior judge is being trained by a senior judge to learn how to assess ice skaters. After the training, the judges each assess 6 ice skaters \(A , B , C , D , E\) and \(F\). They each list them in order of preference with the best ice skater first. The results are shown in the table below.

1. Calculate Spearman's rank correlation coefficient for these data.
2. Test, at the \(5 \%\) level of significance, whether or not there is evidence of a positive correlation between the rankings of the junior judge and the senior judge. State your hypotheses clearly.
3. Comment on the effectiveness of the training delivered by the senior judge.

4. A psychologist carries out a survey of the perceived body weight of 150 randomly chosen people. He asks them if they think they are underweight, about right or overweight. His results are summarised in the table below. The psychologist calculates two of the expected frequencies, to 2 decimal places, for a test of independence between perceived body weight and gender. These results are shown in the table below.

1. Complete the table of expected frequencies shown above.
2. Test, at the \(10 \%\) level of significance, whether or not perceived body weight is independent of gender. State your hypotheses clearly. The psychologist now combines the male and female data to test whether or not body weight types are chosen equally.
3. Find the smallest significance level, from the tables in the formula booklet, for which there is evidence of a preference.

5. Paul takes the company bus to work. According to the bus timetable he should arrive at work at 0831. Paul believes the bus is not reliable and often arrives late. Paul decides to test the arrival time of the bus and carries out a survey. He records the values of the random variable $$X = \text { number of minutes after } 0831 \text { when the bus arrives. }$$ His results are summarised below. $$n = 15 \quad \sum x = 60 \quad \sum x ^ { 2 } = 1946$$

1. Calculate unbiased estimates of the mean, \(\mu\), and the variance of \(X\). Using the mean of Paul's sample and given \(X \sim \mathrm {~N} \left( \mu , 10 ^ { 2 } \right)\)
2. 1. calculate a 95\% confidence interval for the mean arrival time at work for this company bus.
   2. State an assumption you made about the values in the sample obtained by Paul.
3. Comment on Paul's belief. Justify your answer.

6. An engineer has developed a new battery. She claims that the new battery will last more than 8 hours longer, on average, than the old battery. To test the claim, the engineer randomly selects a sample of 50 new batteries and 40 old batteries. She records how long each battery lasts, \(x\) hours for the new batteries and \(y\) hours for the old batteries. The results are summarised in the table below.

1. Test, at the \(5 \%\) level of significance, whether or not there is evidence to support the engineer's claim. State your hypotheses and show your working clearly.
2. Explain the relevance of the Central Limit Theorem to the test in part (a).

7. Sugar is packed into medium bags and large bags. The weights of the medium bags of sugar are normally distributed with mean 520 grams and standard deviation 10 grams. The weights of the large bags of sugar are normally distributed with mean 1510 grams and standard deviation 20 grams.

1. Find the probability that a randomly chosen large bag of sugar weighs at least 15 grams more than the combined weight of 3 randomly chosen medium bags of sugar.
2. Find the probability that a randomly chosen large bag of sugar weighs less than 3 times the weight of a randomly chosen medium bag of sugar. A random sample of 5 medium bags of sugar is taken.
3. Find the value of \(d\) so that the probability that all 5 bags of sugar each weigh more than 520 grams is equal to the probability that the mean weight of the 5 bags of sugar is more than \(d\) grams.