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.

Penshop have stores selling stationary in each of 6 towns. The population, \(P\), in tens of thousands and the monthly turnover, \(T\), in thousands of pounds for each of the shops are as recorded below.

Town	Abberton	Bember	Claster	Deller	Edgeton	Figland
\(P\) (0.000's)	3.2	7.6	5.2	9.0	8.1	4.8
\(T\) (£ 000's)	11.1	12.4	13.3	19.3	17.9	11.8

1. Represent these data on a scatter diagram with \(T\) on the vertical axis. [4]
2. 1. Which town's shop might appear to be underachieving given the populations of the towns?
   2. Suggest two other factors that might affect each shop's turnover. [3]

You may assume that $$\Sigma P = 37.9, \quad \Sigma T = 85.8, \quad \Sigma P^2 = 264.69, \quad \Sigma T^2 = 1286, \quad \Sigma PT = 574.25.$$

1. Find the equation of the regression line of \(T\) on \(P\). [7]
2. Estimate the monthly turnover that might be expected if a shop were opened in Gratton, a town with a population of 68 000. [2]
3. Why might the management of Penshop be reluctant to use the regression line to estimate the monthly turnover they could expect if a shop were opened in Haggin, a town with a population of 172 000? [1]