| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Linear regression |
| Type | Explain least squares concept |
| Difficulty | Standard +0.3 Part (a) is a standard least squares calculation using given summations—routine for Further Maths students. Part (b) tests conceptual understanding of the least squares criterion (straightforward recall). Part (c) requires applying a linear transformation to the regression equation, which is a standard technique but requires careful algebraic manipulation. Overall slightly easier than average A-level difficulty due to the computational nature and standard techniques involved. |
| Spec | 5.09a Dependent/independent variables5.09b Least squares regression: concepts5.09c Calculate regression line5.09d Linear coding: effect on regression5.09e Use regression: for estimation in context |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (a) | y = 52.7 + 0.251x |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | a in range [0.250, 0.251] |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | (b) | This quantity is minimised to find best-fit line |
| [1] | 2.4 | Need “minimised” or “this is its minimum value” OE |
| 1 | (c) | y′ = 11.5 + 0.139x |
| Answer | Marks |
|---|---|
| 9 9 | M1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | Apply inverse formula at least once |
Question 1:
1 | (a) | y = 52.7 + 0.251x | B1*
B1*
depB1
[3] | 1.1
1.1
1.1 | a in range [0.250, 0.251]
b correct to 3 SF
Completely correct including letters
SC: Correct formulae used for a and b: M1(A1)A1
1 | (b) | This quantity is minimised to find best-fit line | B1
[1] | 2.4 | Need “minimised” or “this is its minimum value” OE
1 | (c) | y′ = 11.5 + 0.139x
[y′ = 5× (their a – 32) + 5× their b]
9 9 | M1
A1ft
[2] | 1.1
1.1 | Apply inverse formula at least once
All correct, any letters, ft on their y1 At a seaside resort the number $X$ of ice-creams sold and the temperature $Y ^ { \circ } \mathrm { F }$ were recorded on 20 randomly chosen summer days. The data can be summarised as follows.\\
$\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297$
\begin{enumerate}[label=(\alph*)]
\item Calculate the equation of the least squares regression line of $y$ on $x$, giving your answer in the form $y = a + b x$.
\item Explain the significance for the regression line of the quantity $\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }$.
\item It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as $f ^ { \circ }$ Fahrenheit and $c ^ { \circ }$ Centigrade, the relationship between $f$ and $c$ is $\mathrm { c } = \frac { 5 } { 9 } ( \mathrm { f } - 32 )$.
Find the equation of the new regression line.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2021 Q1 [6]}}