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. 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. $p=a+b\ln m \qquad p=d+e\sqrt{m}$ 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] 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. Draw the line $y=\frac{5}{2}n+1$ on the scatter diagram above.[1] For the model $y=\mathrm{f}(n)$, the residual for a point $(a,b)$ is $b-\mathrm{f}(a)$. Mark the residuals for the points on the scatter diagram above.[1] 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] 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. 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] 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]

N2024 P2 Q11

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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]

\(m\)	\(24\)	\(250\)	\(550\)	\(1000\)	\(2000\)	\(5000\)
\(p\)	\(100\)	\(300\)	\(400\)	\(600\)	\(900\)	\(1200\)

\(n\)	\(2\)	\(5\)	\(6\)	\(7\)	\(11\)
\(y\)	\(6\)	\(10\)	\(16\)	\(20\)	\(25\)

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