| Exam Board | OCR |
|---|---|
| Module | Further Statistics AS (Further Statistics AS) |
| Year | 2024 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Bivariate data |
| Type | Calculate r from summary statistics |
| Difficulty | Standard +0.3 This is a standard Further Statistics question requiring routine application of correlation and regression formulas from summary statistics. Parts (a)-(d) involve direct substitution into standard formulas, part (e) tests understanding of scale transformations, and part (f) requires basic interpretation. While it's a multi-part question worth several marks, each component is straightforward with no novel problem-solving required—slightly easier than average even for Further Maths. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.836\ (0.835657)\) | B2 [2] | Allow \([0.835, 0.836]\). If B0, allow B1 if two of 366, 0.0402, 3.21 or two of 5494(.38), 0.604, 48.14 seen, or correct substitution into formula |
| Answer | Marks | Guidance |
|---|---|---|
| \(t = 0.135 + 0.00876x\) or \(t = 0.135 + \frac{3407}{274720}x\) | B2 [2] | \(a\): awrt 0.135. \(b\): exact or in range \([0.0087, 0.0088]\). Needs letters correct. SC: One error: B1 |
| Answer | Marks | Guidance |
|---|---|---|
| The values of \(a\) and \(b\) found in (b) are chosen to minimise this quantity | B1 [1] | Require "minimised" OE. Allow "shows the minimised sum of squares". *Not* "sum of least squares"; *not* just "sum of squares of residuals"; *not* "the lower this value, the better the line of best fit" |
| Answer | Marks | Guidance |
|---|---|---|
| \(0.503\ (0.502897)\) | B1 [1] | Awrt 0.503 |
| Answer | Marks | Guidance |
|---|---|---|
| No effect (or *their* 0.836) | B1ft [1] | Reason not needed, ft on their (a) |
| Answer | Marks | Guidance |
|---|---|---|
| \(t' = 1.35 + 0.0876x\) | B1ft [1] | OE, e.g. "\(a\) and \(b\) multiplied by 10" or "\(t\) becomes \(t/10\)". Ignore letters. Must refer to equation, so *not* "all values ten times greater". "Increase \(a\) and \(b\) by 10": BOD B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Model seems to be based on people whose ages are not close to 42 | B1 | Relevant comment on 42 and data, e.g. "42 is outside the data range(s)". Allow "biased towards old and young people", allow "no one of age close to 42", but *not* "no one of age 42". *Not* "42 is in data range" |
| \(r\) is quite high *or* there is quite strong correlation *or* points are likely to lie fairly close to straight line | B1 | Relevant comment based on value of \(r\). *Not* "the value of \(r\) is unreliable"; *not* "there is significant evidence of correlation". SC: "Data may be clumped so value of \(r\) is uninformative" oe: B1 |
| Reliability dubious, or hard to tell | B1 [3] | Single nuanced conclusion, not inconsistent with previous comments, must have referred to both 42 and pmcc, but needs only 1 of previous 2 marks. Allow "unreliable/not reliable". *Not* "reliable" or stronger, not "slightly unreliable". Allow "probably unreliable" but *not* "might not be reliable" etc. Irrelevant comments max 2/3 if any wrong comments seen |
# Question 3:
## Part (a)
$0.836\ (0.835657)$ | B2 [2] | Allow $[0.835, 0.836]$. If B0, allow B1 if two of 366, 0.0402, 3.21 or two of 5494(.38), 0.604, 48.14 seen, or correct substitution into formula
## Part (b)
$t = 0.135 + 0.00876x$ or $t = 0.135 + \frac{3407}{274720}x$ | B2 [2] | $a$: awrt 0.135. $b$: exact or in range $[0.0087, 0.0088]$. Needs letters correct. SC: One error: B1
## Part (c)
The values of $a$ and $b$ found in **(b)** are chosen to minimise this quantity | B1 [1] | Require "minimised" OE. Allow "shows the minimised sum of squares". *Not* "sum of least squares"; *not* just "sum of squares of residuals"; *not* "the lower this value, the better the line of best fit"
## Part (d)
$0.503\ (0.502897)$ | B1 [1] | Awrt 0.503
## Part (e)(i)
No effect (or *their* 0.836) | B1ft [1] | Reason not needed, ft on their **(a)**
## Part (e)(ii)
$t' = 1.35 + 0.0876x$ | B1ft [1] | OE, e.g. "$a$ and $b$ multiplied by 10" or "$t$ becomes $t/10$". Ignore letters. Must refer to equation, so *not* "all values ten times greater". "Increase $a$ and $b$ by 10": BOD B1
## Part (f)
Model seems to be based on people whose ages are not close to 42 | B1 | Relevant comment on 42 and data, e.g. "42 is outside the data range(s)". Allow "biased towards old and young people", allow "no one of age close to 42", but *not* "no one of age 42". *Not* "42 is in data range"
$r$ is quite high *or* there is quite strong correlation *or* points are likely to lie fairly close to straight line | B1 | Relevant comment based on value of $r$. *Not* "the value of $r$ is unreliable"; *not* "there is significant evidence of correlation". SC: "Data may be clumped so value of $r$ is uninformative" oe: B1
Reliability dubious, or hard to tell | B1 [3] | Single nuanced conclusion, not inconsistent with previous comments, must have referred to both 42 and pmcc, but needs only 1 of previous 2 marks. Allow "unreliable/not reliable". *Not* "reliable" or stronger, not "slightly unreliable". Allow "probably unreliable" but *not* "might not be reliable" etc. Irrelevant comments max 2/3 if any wrong comments seen3 The ages, $x$ years, and the reaction time, $t$ seconds, in an experiment carried out on a sample of 15 volunteers are summarised as follows.\\
$n = 15 \quad \sum x = 762 \quad \sum t = 8.7 \quad \sum x ^ { 2 } = 44204 \quad \sum t ^ { 2 } = 5.65 \quad \sum x t = 490.1$
\begin{enumerate}[label=(\alph*)]
\item Calculate the value of the product moment correlation coefficient between $x$ and $t$.
\item Calculate the equation of the line of regression of $t$ on $x$. Give your answer in the form $\mathrm { t } = \mathrm { a } + \mathrm { bx }$ where $a$ and $b$ are constants to be determined.
\item Explain the relevance of the quantity $\sum ( t - a - b x ) ^ { 2 }$ to your answer to part (b).
\item Estimate the reaction time, in seconds, for a volunteer aged 42.
It is subsequently decided to measure the reaction time in tenths of a second rather than in seconds (so, for example, a time of 0.6 seconds would now be recorded as 6 ).
\item \begin{enumerate}[label=(\roman*)]
\item State what effect, if any, this change would have on your answer to part (a).
\item State what effect, if any, this change would have on your answer to part (b).
It is known that the sample of 15 volunteers consisted almost entirely of students and retired people.
\end{enumerate}\item Using this information, and the value of the product moment correlation coefficient, comment on the reliability of your estimate in part (d).
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics AS 2024 Q3 [11]}}