Draw scatter diagram from data

1 An experiment is performed to determine the response of maize to nitrogen fertilizer. Data for the amount of nitrogen fertilizer applied, \(x \mathrm {~kg} / \mathrm { hectare }\), and the average yield of maize, \(y\) tonnes/hectare, in 5 experimental plots are given in the table below.

1. Draw a scatter diagram to illustrate these data.
2. Calculate the equation of the regression line of \(y\) on \(x\).
3. Draw your regression line on your scatter diagram and comment briefly on its fit.
4. Calculate the value of the residual for the data point where \(x = 30\) and \(y = 2.5\).
5. Use the equation of the regression line to calculate estimates of average yield with nitrogen fertilizer applications of
   (A) \(45 \mathrm {~kg} / \mathrm { hectare }\),
   (B) \(150 \mathrm {~kg} /\) hectare.
6. In a plot where \(150 \mathrm {~kg} /\) hectare of nitrogen fertilizer is applied, the average yield of maize is 8.7 tonnes/hectare. Comment on this result.

3. A manufacturer stores drums of chemicals. During storage, evaporation takes place. A random sample of 10 drums was taken and the time in storage, \(x\) weeks, and the evaporation loss, \(y \mathrm { ml }\), are shown in the table below.

1. On graph paper, draw a scatter diagram to represent these data.
2. Give a reason to support fitting a regression model of the form \(y = a + b x\) to these data.
3. Find, to 2 decimal places, the value of \(a\) and the value of \(b\). $$\text { (You may use } \Sigma x ^ { 2 } = 1352 , \Sigma y ^ { 2 } = 53112 \text { and } \Sigma x y = 8354 \text {.) }$$
4. Give an interpretation of the value of \(b\).
5. Using your model, predict the amount of evaporation that would take place after
   1. 19 weeks,
   2. 35 weeks.
6. Comment, with a reason, on the reliability of each of your predictions.