1 In a science investigation into energy conservation in the home, a student is collecting data on the time taken for an electric kettle to boil as the volume of water in the kettle is varied. The student's data are shown in the table below, where \(v\) litres is the volume of water in the kettle and \(t\) seconds is the time taken for the kettle to boil (starting with the water at room temperature in each case). Also shown are summary statistics and a scatter diagram on which the regression line of \(t\) on \(v\) is drawn.
| \(v\) | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 |
| \(t\) | 44 | 78 | 114 | 156 | 172 |
$$n = 5 , \Sigma v = 3.0 , \Sigma t = 564 , \Sigma v ^ { 2 } = 2.20 , \Sigma v t = 405.2 .$$
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- Calculate the equation of the regression line of \(t\) on \(v\), giving your answer in the form \(t = a + b v\).
- Use this equation to predict the time taken for the kettle to boil when the amount of water which it contains is
(A) 0.5 litres,
(B) 1.5 litres.
Comment on the reliability of each of these predictions. - In the equation of the regression line found in part (i), explain the role of the coefficient of \(v\) in the relationship between time taken and volume of water.
- Calculate the values of the residuals for \(v = 0.8\) and \(v = 1.0\).
- Explain how, on a scatter diagram with the regression line drawn accurately on it, a residual could be measured and its sign determined.
(a) A farmer grows Brussels sprouts. The diameter of sprouts in a particular batch, measured in mm , is Normally distributed with mean 28 and variance 16. Sprouts that are between 24 mm and 33 mm in diameter are sold to a supermarket. - Find the probability that the diameter of a randomly selected sprout will be within this range.
- The farmer sells the sprouts in this range to the supermarket for 10 pence per kilogram. The farmer sells sprouts under 24 mm in diameter to a frozen food factory for 5 pence per kilogram. Sprouts over 33 mm in diameter are thrown away. Estimate the total income received by the farmer for the batch, which weighs 25000 kg .
- By harvesting sprouts earlier, the mean diameter for another batch can be reduced to \(k \mathrm {~mm}\). Find the value of \(k\) for which only \(5 \%\) of the sprouts will be above 33 mm in diameter. You may assume that the variance is still 16 .
(b) The farmer also grows onions. The weight in kilograms of the onions is Normally distributed with mean 0.155 and variance 0.005 . He is trying out a new variety, which he hopes will yield a higher mean weight. In order to test this, he takes a random sample of 25 onions of the new variety and finds that their total weight is 4.77 kg . You should assume that the weight in kilograms of the new variety is Normally distributed with variance 0.005 . - Write down suitable null and alternative hypotheses for the test in terms of \(\mu\). State the meaning of \(\mu\) in this case.
- Carry out the test at the \(1 \%\) level.