| Exam Board | OCR |
|---|---|
| Module | FS1 AS (Further Statistics 1 AS) |
| Year | 2017 |
| Session | December |
| Marks | 8 |
| Topic | Bivariate data |
| Type | Identify outliers or unusual points |
| Difficulty | Moderate -0.5 This is a standard bivariate data question requiring routine application of formulas for correlation coefficient and regression line, with straightforward conceptual questions about dependence and invariance properties. All summations are provided, making calculations mechanical. The 'least squares' explanation is textbook recall. Slightly easier than average due to the computational scaffolding provided. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09e Use regression: for estimation in context |
| \(T\) | 17 | 21 | 25 | 26 | 27 | 27 | 29 | 30 | 30 |
| \(C\) | 21 | 16 | 20 | 38 | 32 | 37 | 35 | 39 | 42 |
| Answer | Marks | Guidance |
|---|---|---|
| \(N\) is likely to be dependent on \(T\) because more ice creams are likely to be sold in hot weather | B1 | oe, any plausible reason that shows understanding of concept |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 0.819\) | B2 | BC: awrt 0.819 |
| SC: answer incorrect, correct working seen: B1 |
| Answer | Marks | Guidance |
|---|---|---|
| No change, \(r\) unchanged by linear coding | B1 | Both "no change" oe and "linear" oe |
| Answer | Marks | Guidance |
|---|---|---|
| \(n = -15.6 + 1.81t\) | B1ft | BC, Numbers correct, to 3 sf |
| dep B1 | Whole equation correct |
| Answer | Marks | Guidance |
|---|---|---|
| Vertical line drawn between line and a point | M1* | At least one line drawn |
| Minimise sum of squares of these lengths | dep E1 | This or equivalent stated |
## (i)
$N$ is likely to be dependent on $T$ because more ice creams are likely to be sold in hot weather | B1 | oe, any plausible reason that shows understanding of concept
## (ii)
$r = 0.819$ | B2 | BC: awrt 0.819
| | SC: answer incorrect, correct working seen: B1
## (iii)
No change, $r$ unchanged by linear coding | B1 | Both "no change" oe and "linear" oe
## (iv)
$n = -15.6 + 1.81t$ | B1ft | BC, Numbers correct, to 3 sf
| dep B1 | Whole equation correct
## (v)
Vertical line drawn between line and a point | M1* | At least one line drawn
Minimise sum of squares of these lengths | dep E1 | This or equivalent stated
---5 A shop manager recorded the maximum daytime temperature $T ^ { \circ } \mathrm { C }$ and the number $C$ of ice creams sold on 9 summer days.
The results are given in the table and illustrated in the scatter diagram.
\begin{center}
\begin{tabular}{ | l | l | l | l | l | l | l | l | l | l | }
\hline
$T$ & 17 & 21 & 25 & 26 & 27 & 27 & 29 & 30 & 30 \\
\hline
$C$ & 21 & 16 & 20 & 38 & 32 & 37 & 35 & 39 & 42 \\
\hline
\end{tabular}
\end{center}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{64d7ed6d-fadd-4c59-afb0-97d1788ba369-3_661_1189_1320_431}
\end{center}
$$n = 9 , \Sigma t = 232 , \Sigma c = 280 , \Sigma t ^ { 2 } = 6130 , \Sigma c ^ { 2 } = 9444 , \Sigma t c = 7489$$
(i) State, with a reason, whether one of the variables $C$ or $T$ is likely to be dependent upon the other.\\
(ii) Calculate Pearson's product-moment correlation coefficient $r$ for the data.\\
(iii) State with a reason what the value of $r$ would have been if the temperature had been measured in ${ } ^ { \circ } \mathrm { F }$ rather than ${ } ^ { \circ } \mathrm { C }$.\\
(iv) Calculate the equation of the least squares regression line of $c$ on $t$.\\
(v) The regression line is drawn on the copy of the scatter diagram in the Printed Answer Booklet. Use this diagram to explain what is meant by "least squares".
\hfill \mbox{\textit{OCR FS1 AS 2017 Q5 [8]}}