Question 2 - A-Level Maths

Edexcel Paper 3 2024 June — Question 2 6 marks

Exam Board	Edexcel
Module	Paper 3 (Paper 3)
Year	2024
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Linear regression
Type	Interpret regression line parameters
Difficulty	Moderate -0.3 This is a straightforward multi-part question on linear regression interpretation requiring standard techniques: interpreting gradient (routine), hypothesis test for correlation with given critical values (textbook procedure), assessing model appropriateness from a scatter diagram (basic interpretation), and finding a constant from context. All parts are standard A-level statistics procedures with no novel problem-solving required, making it slightly easier than average.
Spec	2.02c Scatter diagrams and regression lines 2.02d Informal interpretation of correlation 2.05f Pearson correlation coefficient 2.05g Hypothesis test using Pearson's r

Amar is studying the flight of a bird from its nest.

He measures the bird's height above the ground, \(h\) metres, at time \(t\) seconds for 10 values of \(t\) Amar finds the equation of the regression line for the data to be \(h = 38.6 - 1.28 t\)

Interpret the gradient of this line. The product moment correlation coefficient between \(h\) and \(t\) is - 0.510
Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight.
You should
- state your hypotheses clearly
- use a \(5 \%\) level of significance
- state the critical value used
Jane draws the following scatter diagram for Amar's data. \includegraphics[max width=\textwidth, alt={}, center]{ab7f7951-e6fe-4853-bb69-8016cf3e796c-06_1024_1033_1135_516}
With reference to the scatter diagram, state, giving a reason, whether or not the regression line \(h = 38.6 - 1.28 t\) is an appropriate model for these data. Jane suggests an improved model using the variable \(u = ( t - k ) ^ { 2 }\) where \(k\) is a constant.
She obtains the equation \(h = 38.1 - 0.78 u\)
Choose a suitable value for \(k\) to write Jane's improved model for \(h\) in terms of \(t\) only.

Show mark scheme Show mark scheme source

Question 2:

Part (a)

Answer	Marks	Guidance
Answer	Mark	Guidance
The height (\(h\)) decreases by about \(1.28\) m for each second of the flight	B1	Value can be 1.3 or 1.28 or "just over 1" per sec; must have units "m" and "s"; "descends" implies "height decreases"; condone "decreases by \(-1.28\) m"

Part (b)

Answer	Marks	Guidance
Answer	Mark	Guidance
\(H_0: \rho = 0 \quad H_1: \rho < 0\)	B1	Both hypotheses correct in terms of \(\rho\); accept \(p\) but not \(r\); must be attached to \(H_0\) and \(H_1\)
\([5\%\) 1-tail cv \(=]\ (\pm)\ 0.5494\)	M1	For critical value corresponding to \(H_1\): 1-tail awrt \(\pm 0.549\); if hypotheses in words, can deduce one or two-tail
\([r = -0.510\) not sig\(]\) there is insufficient evidence of a negative correlation between height (or \(h\)) and time (or \(t\))	A1	Correct conclusion mentioning correlation, height and time; "not sig" not needed but if seen must be correct; do NOT award if contradictory comments e.g. "reject \(H_0\)"; comparison of 0.510 with 0.05 or \(-0.549 > -0.510\) scores B0

SC: B0 (for 2-tail) M0 (for cv \(= \pm 0.549\)): allow 1 mark (B0M0A1) for conclusion such as "insufficient evidence of (negative) correlation between height and time of flight"

Part (c)

Answer	Marks	Guidance
Answer	Mark	Guidance
No – since points seem to follow a curve/quadratic (rather than a line); or since points are "non-linear" but regression line/model is linear; or e.g. between (\(t=5\) and \(7\)) height drops by much more than 2.56 m; or e.g. gradient is positive up to \(t=3.5\) (line gradient \(< 0\)); or e.g. gradient is positive initially (line gradient \(< 0\)); or e.g. gradient is positive and then negative	B1	For saying no AND giving a suitable supporting reason; don't allow "correlation" on its own instead of "gradient"; B0 for "points don't lie close to a straight line" – need mention of curve or other feature differing from regression line; B0 for just "non-linear" without mention of model being linear; B0 for simply comparing 1 or 2 points

Part (d)

Answer	Marks	Guidance
Answer	Mark	Guidance
\([h = 38.1 - 0.78(t-k)^2\) with\(]\) a suitable \(k\) i.e. in the range \(3 \sim 4.5\)	B1	For a value of \(k\) in the range \([3, 4.5]\); do not need \(k = \ldots\); accept a value embedded in Jane's model; ISW any errors in multiplying out bracket

# Question 2:

## Part (a)
| Answer | Mark | Guidance |
|--------|------|----------|
| The height ($h$) decreases by about $1.28$ m for each second of the flight | B1 | Value can be 1.3 or 1.28 or "just over 1" per sec; must have units "m" and "s"; "descends" implies "height decreases"; condone "decreases by $-1.28$ m" |

## Part (b)
| Answer | Mark | Guidance |
|--------|------|----------|
| $H_0: \rho = 0 \quad H_1: \rho < 0$ | B1 | Both hypotheses correct in terms of $\rho$; accept $p$ but not $r$; must be attached to $H_0$ and $H_1$ |
| $[5\%$ 1-tail cv $=]\ (\pm)\ 0.5494$ | M1 | For critical value corresponding to $H_1$: 1-tail awrt $\pm 0.549$; if hypotheses in words, can deduce one or two-tail |
| $[r = -0.510$ not sig$]$ there is insufficient evidence of a negative correlation between height (or $h$) and time (or $t$) | A1 | Correct conclusion mentioning correlation, height and time; "not sig" not needed but if seen must be correct; do NOT award if contradictory comments e.g. "reject $H_0$"; comparison of 0.510 with 0.05 or $-0.549 > -0.510$ scores B0 |

**SC:** B0 (for 2-tail) M0 (for cv $= \pm 0.549$): allow 1 mark (B0M0A1) for conclusion such as "insufficient evidence of (negative) correlation between height and time of flight"

## Part (c)
| Answer | Mark | Guidance |
|--------|------|----------|
| No – since points seem to follow a curve/quadratic (rather than a line); or since points are "non-linear" but regression line/model is linear; or e.g. between ($t=5$ and $7$) height drops by much more than 2.56 m; or e.g. gradient is positive up to $t=3.5$ (line gradient $< 0$); or e.g. gradient is positive initially (line gradient $< 0$); or e.g. gradient is positive and then negative | B1 | For saying no AND giving a suitable supporting reason; don't allow "correlation" on its own instead of "gradient"; B0 for "points don't lie close to a straight line" – need mention of curve or other feature differing from regression line; B0 for just "non-linear" without mention of model being linear; B0 for simply comparing 1 or 2 points |

## Part (d)
| Answer | Mark | Guidance |
|--------|------|----------|
| $[h = 38.1 - 0.78(t-k)^2$ with$]$ a suitable $k$ i.e. in the range $3 \sim 4.5$ | B1 | For a value of $k$ in the range $[3, 4.5]$; do not need $k = \ldots$; accept a value embedded in Jane's model; ISW any errors in multiplying out bracket |

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Show LaTeX source

\begin{enumerate}
  \item Amar is studying the flight of a bird from its nest.
\end{enumerate}

He measures the bird's height above the ground, $h$ metres, at time $t$ seconds for 10 values of $t$\\
Amar finds the equation of the regression line for the data to be $h = 38.6 - 1.28 t$\\
(a) Interpret the gradient of this line.

The product moment correlation coefficient between $h$ and $t$ is - 0.510\\
(b) Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight.\\
You should

\begin{itemize}
  \item state your hypotheses clearly
  \item use a $5 \%$ level of significance
  \item state the critical value used
\end{itemize}

Jane draws the following scatter diagram for Amar's data.\\
\includegraphics[max width=\textwidth, alt={}, center]{ab7f7951-e6fe-4853-bb69-8016cf3e796c-06_1024_1033_1135_516}\\
(c) With reference to the scatter diagram, state, giving a reason, whether or not the regression line $h = 38.6 - 1.28 t$ is an appropriate model for these data.

Jane suggests an improved model using the variable $u = ( t - k ) ^ { 2 }$ where $k$ is a constant.\\
She obtains the equation $h = 38.1 - 0.78 u$\\
(d) Choose a suitable value for $k$ to write Jane's improved model for $h$ in terms of $t$ only.

\hfill \mbox{\textit{Edexcel Paper 3 2024 Q2 [6]}}

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