AQA S1 (Statistics 1) 2016 June

1 The table shows the heights, \(x \mathrm {~cm}\), and the arm spans, \(y \mathrm {~cm}\), of a random sample of 12 men aged between 21 years and 40 years.

1. Calculate the value of the product moment correlation coefficient between \(x\) and \(y\).
2. Interpret, in context, your value calculated in part (a).

2 A small chapel was open to visitors for 55 days during the summer of 2015. The table summarises the daily numbers of visitors.

1. For these data:
   1. state the modal value;
   2. find values for the median and the interquartile range.
2. Name one measure of average and one measure of spread that cannot be calculated exactly from the data in the table.
   [0pt] [2 marks]
3. Reference to the raw data revealed that the 3 unknown exact values in the table were 13,37 and 58. Making use of this additional information, together with the data in the table, calculate the value of each of the two measures that you named in part (b).
   [0pt] [3 marks]

3 The table shows, for a random sample of 500 patients attending a dental surgery, the patients' ages, in years, and the NHS charge bands for the patients' courses of treatment. Band 0 denotes the least expensive charge band and band 3 denotes the most expensive charge band.

1. Calculate, to three decimal places, the probability that a patient, selected at random from these 500 patients, was:
   1. aged between 41 and 65;
   2. aged 66 or over and charged at band 2;
   3. aged between 19 and 40 and charged at most at band 1;
   4. aged 41 or over, given that the patient was charged at band 2;
   5. charged at least at band 2, given that the patient was not aged 66 or over.
2. Four patients at this dental surgery, not included in the above 500 patients, are selected at random. Estimate, to three significant figures, the probability that two of these four patients are aged between 41 and 65 and are not charged at band 0 , and the other two patients are aged 66 or over and are charged at either band 1 or band 2.
   [0pt] [5 marks]

4 As part of her science project, a student found the mass, \(y\) grams, of a particular compound that dissolved in 100 ml of water at each of 12 different set temperatures, \(x ^ { \circ } \mathrm { C }\). The results are shown in the table.

1. Calculate the equation of the least squares regression line of \(y\) on \(x\).
2. Interpret, in context, your value for the gradient of this regression line.
3. Use your equation to estimate the mass of the compound which will dissolve in 100 ml of water at \(68 ^ { \circ } \mathrm { C }\).
4. Given that the values of the 12 residuals for the regression line of \(y\) on \(x\) lie between - 7 and + 9 , comment, with justification, on the likely accuracy of your estimate in part (c).
   [0pt] [2 marks]

5 Still mineral water is supplied in 1.5-litre bottles. The actual volume, \(X\) millilitres, in a bottle may be modelled by a normal distribution with mean \(\mu = 1525\) and standard deviation \(\sigma = 9.6\).

1. Determine the probability that the volume of water in a randomly selected bottle is:
   1. less than 1540 ml ;
   2. more than 1535 ml ;
   3. between 1515 ml and 1540 ml ;
   4. not 1500 ml .
2. The supplier requires that only 10 per cent of bottles should contain more than 1535 ml of water. Assuming that there has been no change in the value of \(\sigma\), calculate the reduction in the value of \(\mu\) in order to satisfy this requirement. Give your answer to one decimal place.
3. Sparkling spring water is supplied in packs of six 0.5 -litre bottles. The actual volume in a bottle may be modelled by a normal distribution with mean 508.5 ml and standard deviation 3.5 ml . Stating a necessary assumption, determine the probability that:
   1. the volume of water in each of the 6 bottles from a randomly selected pack is more than 505 ml ;
   2. the mean volume of water in the 6 bottles from a randomly selected pack is more than 505 ml .
      [0pt] [7 marks]

6 The proportions of different colours of loom bands in a box of 10000 loom bands are given in the table.

1. A sample of 50 loom bands is selected at random from the box. Use a binomial distribution with \(n = 50\), together with relevant information from the table, to estimate the probability that this sample contains:
   1. exactly 4 red loom bands;
   2. at most 10 yellow loom bands;
   3. at least 30 blue or green loom bands;
   4. more than 35 but fewer than 45 loom bands that are neither yellow nor white.
2. The random variable \(R\) denotes the number of red loom bands in a random sample of \(\mathbf { 3 0 0 }\) loom bands selected from the box. Estimate values for the mean and the variance of \(R\).
   [0pt] [2 marks]

7 Customers buying euros ( €) at a travel agency must pay for them in pounds ( \(\pounds\) ). The amounts paid, \(\pounds x\), by a sample of 40 customers were, in ascending order, as follows.