A certain brand of breakfast cereal is sold in a variety of different sized packs. Details of the mass of cereal in each pack, \(m\) grams, and the price, \(p\) cents, are given in the table below.
\(m\) \(24\) \(250\) \(550\) \(1000\) \(2000\) \(5000\)
\(p\) \(100\) \(300\) \(400\) \(600\) \(900\) \(1200\)
A scatter diagram for the data is shown below.
1. Explain what the scatter diagram tells you about the relationship between \(m\) and \(p\).[1]
The following two models are proposed, where \(a\), \(b\), \(d\), and \(e\) are constants.
\(p=a+b\ln m \qquad p=d+e\sqrt{m}\)
1. Determine which of these models gives the better fit to the data. State the values of the constants and the product moment correlation coefficient in this case.[4]
2. A new pack containing \(750\) grams of cereal is introduced. Use the model you identified in part (a)(ii) to estimate the price of this pack, correct to the nearest \(10\) cents. Explain whether your answer is reliable.
  [2]
Kai is investigating the relationship between the number, \(n\), of a particular type of high-performance batteries in a pack and the price, \(\$y\), of the pack she found in different stores. Her results are shown in the table below.
\(n\) \(2\) \(5\) \(6\) \(7\) \(11\)
\(y\) \(6\) \(10\) \(16\) \(20\) \(25\)
This information is shown in the scatter diagram below.
Kai decides to investigate whether the model \(y=\frac{5}{2}n+1\) is a good fit for this data.
1. Draw the line \(y=\frac{5}{2}n+1\) on the scatter diagram above.[1]
1. For the model \(y=\mathrm{f}(n)\), the residual for a point \((a,b)\) is \(b-\mathrm{f}(a)\).
  1. Mark the residuals for the points on the scatter diagram above.[1]
  2. Explain why Kai should use the sum of the squares of the residuals rather than the sum of the residuals when assessing the fit of the model.[1]
  3. Calculate the sum of the squares of the residuals for the line \(y=\frac{5}{2}n+1\).[1]
Kai’s friend points out that other lines parallel to Kai’s line can be drawn which are a better fit for the data.
1. Explain how the sum of the squares of the residuals for a line that is a better fit for the data differs from the sum of the squares of the residuals for the line \(y=\frac{5}{2}n+1\) found in part (ii)(C).
  [1]
2. Find the range of values of \(c\) for which the line \(y=\frac{5}{2}n+c\) is a better fit for the data than the line \(y=\frac{5}{2}n+1\).[3]