Edexcel S3 (Statistics 3) 2016 June

1. (a) State two reasons why stratified sampling might be a more suitable sampling method than simple random sampling.
   (b) State two reasons why stratified sampling might be a more suitable sampling method than quota sampling.
2. A new drug to vaccinate against influenza was given to 110 randomly chosen volunteers. The volunteers were given the drug in one of 3 different concentrations, \(A , B\) and \(C\), and then were monitored to see if they caught influenza. The results are shown in the table below.

Test, at the \(10 \%\) level of significance, whether or not there is an association between catching influenza and the concentration of the new drug. State your hypotheses and show your working clearly. You should state your expected frequencies to 2 decimal places.
(10)

3. (a) Describe when you would use Spearman's rank correlation coefficient rather than the product moment correlation coefficient to measure the strength of the relationship between two variables.
(1) A shop sells sunglasses and ice cream. For one week in the summer the shopkeeper ranked the daily sales of ice cream and sunglasses. The ranks are shown in the table below. (b) Calculate Spearman's rank correlation coefficient for these data.
(c) Test, at the \(5 \%\) level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses. State your hypotheses clearly. The shopkeeper calculates the product moment correlation coefficient from his raw data and finds \(r = 0.65\)
(d) Using this new coefficient, test, at the \(5 \%\) level of significance, whether or not there is a positive correlation between sales of ice cream and sales of sunglasses.
(e) Using your answers to part (c) and part (d), comment on the nature of the relationship between sales of sunglasses and sales of ice cream.

4. The weights of eggs are normally distributed with mean 60 g and standard deviation 5 g Sairah chooses 2 eggs at random.

1. Find the probability that the difference in weight of these 2 eggs is more than 2 g
   (5) Sairah is packing eggs into cartons. The weight of an empty egg carton is normally distributed with mean 40 g and standard deviation 1.5 g
2. Find the distribution of the total weight of a carton filled with 12 randomly chosen eggs.
3. Find the probability that a randomly chosen carton, filled with 12 randomly chosen eggs, weighs more than 800 g
   

5. A doctor claims there is a higher mean lung capacity in people who exercise regularly compared to people who do not exercise regularly. He measures the lung capacity, \(x\), of 35 people who exercise regularly and 42 people who do not exercise regularly. His results are summarised in the table below.

1. Test, at the \(5 \%\) level of significance, the doctor's claim. State your hypotheses clearly.
2. State any assumptions you have made in testing the doctor's claim. The doctor decides to add another person who exercises regularly to his data. He measures the person's lung capacity and finds \(x = 31.7\)
3. Find the unbiased estimate of the variance for the sample of 36 people who exercise regularly. Give your answer to 3 significant figures.

6. An airport manager carries out a survey of families and their luggage. Each family is allowed to check in a maximum of 4 suitcases. She observes 50 families at the check-in desk and counts the total number of suitcases each family checks in. The data are summarised in the table below. The manager claims that the data can be modelled by a binomial distribution with \(p = 0.3\)

1. Test the manager's claim at the \(5 \%\) level of significance. State your hypotheses clearly.
   Show your working clearly and give your expected frequencies to 2 decimal places.
   (8) The manager also carries out a survey of the time taken by passengers to check in. She records the number of passengers that check in during each of 100 five-minute intervals. The manager makes a new claim that these data can be modelled by a Poisson distribution. She calculates the expected frequencies given in the table below.

   Number of passengers	0	1	2	3	4	5 or more
   Observed frequency	5	40	31	18	6	0
   Expected frequency	16.53	29.75	\(r\)	\(s\)	7.23	3.64

2. Find the value of \(r\) and the value of \(s\) giving your answers to 2 decimal places.
3. Stating your hypotheses clearly, use a \(1 \%\) level of significance to test the manager's new claim.

7. A restaurant states that its hamburgers contain \(20 \%\) fat. Paul claims that the mean fat content of their hamburgers is less than \(20 \%\). Paul takes a random sample of 50 hamburgers from the restaurant and finds that they contain a mean fat content of 19.5\% with a standard deviation of 1.5\% You may assume that the fat content of hamburgers is normally distributed.

1. Find the \(90 \%\) confidence interval for the mean fat content of hamburgers from the restaurant.
2. State, with a reason, what action Paul should recommend the restaurant takes over the stated fat content of their hamburgers. The restaurant changes the mean fat content of their hamburgers to \(\mu \%\) and adjusts the standard deviation to \(2 \%\). Paul takes a sample of size \(n\) from this new batch of hamburgers. He uses the sample mean \(\bar { X }\) as an estimator of \(\mu\).
3. Find the minimum value of \(n\) such that \(\mathrm { P } ( | \bar { X } - \mu | < 0.5 ) \geqslant 0.9\)