6 A meteorologist is investigating the relationship between altitude \(x\) metres and mean annual temperature \(y ^ { \circ } \mathrm { C }\) in an American state.
She selects 12 locations at various altitudes and then stations a remote monitoring device at each of them to measure the temperature over the course of a year. Fig. 6 illustrates the data which she obtains.
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\caption{Fig. 6}
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- Explain why it would not be appropriate to carry out a hypothesis test for correlation based on the product moment correlation coefficient.
- Explain why altitude has been plotted on the horizontal axis in Fig. 6.
Summary statistics for \(x\) and \(y\) are as follows.
$$\sum x = 21200 \quad \sum y = 105.4 \quad \sum x ^ { 2 } = 39100000 \quad \sum y ^ { 2 } = 1004 \quad \sum x y = 176090$$
- Calculate the equation of the regression line of \(y\) on \(x\).
- Use the equation of the regression line to predict the values of the mean annual temperature at each of the following altitudes.
- 2000 metres
- 3000 metres
- Comment on the reliability of your predictions in part (d).
- Calculate the value of the residual for the data point ( \(1600,8.1\) ).