| Exam Board | OCR |
|---|---|
| Module | Further Statistics (Further Statistics) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Interpret PMCC value contextually |
| Difficulty | Standard +0.3 This is a straightforward hypothesis testing question on correlation with standard calculations. Part (a)(i) requires interpreting a correlation coefficient (basic recall), part (a)(ii) is a routine one-tailed correlation test following a standard procedure, and part (b) involves recalculating the correlation coefficient after removing one data point using given summaries. All techniques are standard A-level Further Statistics with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Book | A | B | C | D | E | F | G | H | I | J | K | L |
| \(x\) | 0.95 | 0.65 | 0.70 | 0.90 | 0.55 | 1.40 | 1.50 | 0.50 | 1.15 | 0.35 | 0.20 | 0.35 |
| \(y\) | 6.06 | 7.00 | 2.00 | 5.87 | 4.00 | 5.36 | 7.19 | 2.50 | 3.00 | 8.29 | 1.37 | 2.00 |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (a) | (i) |
| [1] | 1.1 | Or equivalent. Must refer to |
| diagram, not just to “correlation” | Ignore extras unless wrong | |
| (ii) | H : ρ = 0, H : ρ > 0, where ρ is the population |
| Answer | Marks |
|---|---|
| correlation between prices in the two years. | B2 |
| Answer | Marks |
|---|---|
| [5] | 1.1 |
| Answer | Marks |
|---|---|
| 2.2b | One error, e.g. ρ not defined, B1 |
| Answer | Marks |
|---|---|
| In context, not too definite | H : no correlation, |
| Answer | Marks |
|---|---|
| Exx | α: Insufficient evidence to reject H . No correlation between … M1A1 (bod) |
| Answer | Marks | Guidance |
|---|---|---|
| (b) | 0.650 | B2 |
| [2] | 3.1a | |
| 1.1 | Full marks for correct answer by any | |
| method | SC: if B0 allow B1 for any 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 2:
2 | (a) | (i) | The points do not lie very close to a straight line | B1
[1] | 1.1 | Or equivalent. Must refer to
diagram, not just to “correlation” | Ignore extras unless wrong
(ii) | H : ρ = 0, H : ρ > 0, where ρ is the population
0 1
pmcc between prices in 1972 and prices in 2018
0.381 < 0.4973
Do not reject H .
0
There is insufficient evidence of (positive)
correlation between prices in the two years. | B2
M1
M1ft
A1ft
[5] | 1.1
2.5
1.1
1.1
2.2b | One error, e.g. ρ not defined, B1
(but allow “population” not stated)
H : r = 0, H : r > 0: same scheme,
0 1
but B2 needs “population” pmcc
Compare with 0.497(3)
Correct first conclusion, needs like-
with-like
In context, not too definite | H : no correlation,
0
H : positive correlation: B1
1
FT on CV 0.5760 only
Exx | α: Insufficient evidence to reject H . No correlation between … M1A1 (bod)
0
β: Wrong first conclusion, correct interpretation: M0A0
γ: Hypotheses wrong way round: maximum M1M1
(b) | 0.650 | B2
[2] | 3.1a
1.1 | Full marks for correct answer by any
method | SC: if B0 allow B1 for any 3
of 8.85, 46.35, 8.8725,
241.7331, 43.153
Question | Answer | Marks | AO | GuidanceA book collector compared the prices of some books, $£x$, when new in 1972 and the prices of copies of the same books, $£y$, on a second-hand website in 2018.
The results are shown in Table 1 and are summarised below the table.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Book & A & B & C & D & E & F & G & H & I & J & K & L \\
\hline
$x$ & 0.95 & 0.65 & 0.70 & 0.90 & 0.55 & 1.40 & 1.50 & 0.50 & 1.15 & 0.35 & 0.20 & 0.35 \\
\hline
$y$ & 6.06 & 7.00 & 2.00 & 5.87 & 4.00 & 5.36 & 7.19 & 2.50 & 3.00 & 8.29 & 1.37 & 2.00 \\
\hline
\end{tabular}
Table 1
$n = 12, \Sigma x = 9.20, \Sigma y = 54.64, \Sigma x^2 = 8.9950, \Sigma y^2 = 310.4572, \Sigma xy = 46.0545$
\begin{enumerate}[label=(\alph*)]
\item It is given that the value of Pearson's product-moment correlation coefficient for the data is 0.381, correct to 3 significant figures.
\begin{enumerate}[label=(\roman*)]
\item State what this information tells you about a scatter diagram illustrating the data. [1]
\item Test at the 5\% significance level whether there is evidence of positive correlation between prices in 1972 and prices in 2018. [5]
\end{enumerate}
\item The collector noticed that the second-hand copy of book J was unusually expensive and he decided to ignore the data for book J.
Calculate the value of Pearson's product-moment correlation coefficient for the other 11 books. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Statistics 2020 Q2 [8]}}