Question 1 - A-Level Maths

OCR MEI S2 2015 June — Question 1 17 marks

Exam Board	OCR MEI
Module	S2 (Statistics 2)
Year	2015
Session	June
Marks	17
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear regression
Type	Calculate y on x from raw data table
Difficulty	Moderate -0.5 This is a straightforward linear regression question requiring standard calculations (means, Sxx, Sxy, regression equation) from a small data table, plus basic residual calculation and interpretation. The only slight variation is part (v) with the regression through origin formula, but this is given explicitly. Below average difficulty as it's purely procedural with no conceptual challenges or problem-solving required.
Spec	2.02c Scatter diagrams and regression lines 5.09a Dependent/independent variables 5.09b Least squares regression: concepts 5.09c Calculate regression line 5.09d Linear coding: effect on regression 5.09e Use regression: for estimation in context

1 A random sample of wheat seedlings is planted and their growth is measured. The table shows their average growth, \(y \mathrm {~mm}\), at half-day intervals.

Time \(t\) days	0	0.5	1	1.5	2	2.5	3
Average growth \(y \mathrm {~mm}\)	0	7	21	33	45	56	62

Draw a scatter diagram to illustrate these data.
Calculate the equation of the regression line of \(y\) on \(t\).
Calculate the value of the residual for the data point at which \(t = 2\).
Use the equation of the regression line to calculate an estimate of the average growth after 5 days for wheat seedlings. Comment on the reliability of this estimate. It is suggested that it would be better to replace the regression line by a line which passes through the origin. You are given that the equation of such a line is \(y = a t\), where \(a = \frac { \sum y t } { \sum t ^ { 2 } }\).
Find the equation of this line and plot the line on your scatter diagram.

Show mark scheme Show mark scheme source

Question 1:

Part (i)

Answer	Marks	Guidance
Answer	Marks	Guidance
Both axes labeled (allow \(t\) and \(y\)) with indication of scale	G1*	Allow axes interchanged; condone \(x\) for \(t\)
Values of time plotted; BOD if \((0,0)\) not clearly visible	G1dep*	(evenly spaced)
Values of average growth plotted; BOD if \((0,0)\) not clearly visible	G1dep*	Visually correct; SC1 for points having correct distribution and G0* awarded; line through origin rewarded in part (v)

Part (ii)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\bar{t} = 1.5,\ \bar{y} = 32\)	B1	For \(\bar{t}\) and \(\bar{y}\) seen or implied by final answer; seen either in calculating \(b\) or forming equation
\(b = \dfrac{S_{yt}}{S_{tt}} = \dfrac{490-(224\times10.5/7)}{22.75-10.5^2/7} = \dfrac{154}{7} = 22\) OR \(b = \dfrac{490/7-(32\times1.5)}{22.75/7-1.5^2} = \dfrac{22}{1} = 22\)	M1	For attempt at gradient \(b\); correct structure needed; FT their \(\bar{t}\) and \(\bar{y}\) for M1
For \(22\)	A1	CAO
\(y - \bar{y} = b(t - \bar{t})\)	M1	For equation of line; with their \(b > 0\), \(\bar{t}\) and \(\bar{y}\)
\(\Rightarrow y - 32 = 22(t-1.5)\) \(\Rightarrow y = 22t - 1\)	A1	CAO; A0 for \(y = 22x - 1\)

Part (iii)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(t=2 \Rightarrow\) predicted average growth \(= (22\times2)-1 = 43\)	B1	For prediction; FT their equation
Residual \(= 45 - 43 = 2\)	M1, A1	For subtraction (either way); FT; \(45 -\) their prediction

Part (iv)

Answer	Marks	Guidance
Answer	Marks	Guidance
\((22\times5)-1 = 109\)	B1	Estimate calculated using equation; FT their equation
Likely to be unreliable as extrapolation (oe)	B1

Part (v)

Answer	Marks	Guidance
Answer	Marks	Guidance
\(a = \dfrac{490}{22.75} = 21.538\ldots = 21.5\) (3 s.f.)	M1, A1
Equation is \(y = 21.5t\)	A1	Allow \(y = 21.54t\) CAO; A0 if axes not scaled or \(a \neq 21.5\) to 3 sf; allow \(y = (280/13)t\)
Line correctly plotted on diagram	A1	Through \((0,0)\) and between \((3,64)\) and \((3,65)\)

# Question 1:

## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Both axes labeled (allow $t$ and $y$) with indication of scale | G1* | Allow axes interchanged; condone $x$ for $t$ |
| Values of time plotted; BOD if $(0,0)$ not clearly visible | G1dep* | (evenly spaced) |
| Values of average growth plotted; BOD if $(0,0)$ not clearly visible | G1dep* | Visually correct; SC1 for points having correct distribution and G0* awarded; line through origin rewarded in part (v) |

## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\bar{t} = 1.5,\ \bar{y} = 32$ | B1 | For $\bar{t}$ and $\bar{y}$ seen or implied by final answer; seen either in calculating $b$ or forming equation |
| $b = \dfrac{S_{yt}}{S_{tt}} = \dfrac{490-(224\times10.5/7)}{22.75-10.5^2/7} = \dfrac{154}{7} = 22$ OR $b = \dfrac{490/7-(32\times1.5)}{22.75/7-1.5^2} = \dfrac{22}{1} = 22$ | M1 | For attempt at gradient $b$; correct structure needed; FT their $\bar{t}$ and $\bar{y}$ for M1 |
| For $22$ | A1 | CAO |
| $y - \bar{y} = b(t - \bar{t})$ | M1 | For equation of line; with their $b > 0$, $\bar{t}$ and $\bar{y}$ |
| $\Rightarrow y - 32 = 22(t-1.5)$ $\Rightarrow y = 22t - 1$ | A1 | CAO; A0 for $y = 22x - 1$ |

## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $t=2 \Rightarrow$ predicted average growth $= (22\times2)-1 = 43$ | B1 | For prediction; FT their equation |
| Residual $= 45 - 43 = 2$ | M1, A1 | For subtraction (either way); FT; $45 -$ their prediction |

## Part (iv)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(22\times5)-1 = 109$ | B1 | Estimate calculated using equation; FT their equation |
| Likely to be **unreliable as extrapolation** (oe) | B1 | |

## Part (v)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $a = \dfrac{490}{22.75} = 21.538\ldots = 21.5$ (3 s.f.) | M1, A1 | |
| Equation is $y = 21.5t$ | A1 | Allow $y = 21.54t$ CAO; A0 if axes not scaled or $a \neq 21.5$ to 3 sf; allow $y = (280/13)t$ |
| Line correctly plotted on diagram | A1 | Through $(0,0)$ and between $(3,64)$ and $(3,65)$ |

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Show LaTeX source

1 A random sample of wheat seedlings is planted and their growth is measured. The table shows their average growth, $y \mathrm {~mm}$, at half-day intervals.

\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | c | c | }
\hline
Time $t$ days & 0 & 0.5 & 1 & 1.5 & 2 & 2.5 & 3 \\
\hline
Average growth $y \mathrm {~mm}$ & 0 & 7 & 21 & 33 & 45 & 56 & 62 \\
\hline
\end{tabular}
\end{center}

(i) Draw a scatter diagram to illustrate these data.\\
(ii) Calculate the equation of the regression line of $y$ on $t$.\\
(iii) Calculate the value of the residual for the data point at which $t = 2$.\\
(iv) Use the equation of the regression line to calculate an estimate of the average growth after 5 days for wheat seedlings. Comment on the reliability of this estimate.

It is suggested that it would be better to replace the regression line by a line which passes through the origin. You are given that the equation of such a line is $y = a t$, where $a = \frac { \sum y t } { \sum t ^ { 2 } }$.\\
(v) Find the equation of this line and plot the line on your scatter diagram.

\hfill \mbox{\textit{OCR MEI S2 2015 Q1 [17]}}

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