| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Minor (Further Statistics Minor) |
| Year | 2020 |
| Session | November |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Spearman’s rank correlation coefficien |
| Type | Hypothesis test for association |
| Difficulty | Moderate -0.3 This is a straightforward application of standard hypothesis testing procedures for Spearman's rank correlation. Parts (a)-(c) involve recognizing when to use Spearman's over Pearson's (routine), calculating the coefficient (mechanical with small n=8), and conducting a standard significance test. Parts (d)-(f) are basic regression calculations using a calculator. All steps are textbook procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 5.08e Spearman rank correlation5.08f Hypothesis test: Spearman rank5.08g Compare: Pearson vs Spearman |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | Because (the grouping of points on) the scatter diagram |
| Answer | Marks |
|---|---|
| due to the outlier/anomalous point | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 3.5a | |
| 2.4 | Or: not bivariate Normal | |
| 5 | (b) | Rank BCG 1 3 6 2 7 5 4 8 |
| Answer | Marks |
|---|---|
| 17 | M1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks |
|---|---|
| 1.1 | For ranking BCG & Measles |
| Answer | Marks |
|---|---|
| BC | Use of percentage values |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (c) | 21 |
| Answer | Marks |
|---|---|
| and measles (in the population) | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.3 |
| Answer | Marks |
|---|---|
| 2.2b | Need to see population in one or |
| Answer | Marks | Guidance |
|---|---|---|
| value, | r | < 1 and conclusion correct |
| Answer | Marks |
|---|---|
| Not too assertive and includes context | SC if outlier excluded in |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (d) | y = -4.80x + 24.49 awrt |
| x = doctors, y = unemployment | B1 |
| Answer | Marks |
|---|---|
| [2] | 1.1 |
| 1.1 | BC |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (e) | 14.89 [14.8, 15.0] |
| [1] | 1.1 | |
| 5 | (f) | Not reliable since extrapolation oe |
| [1] | 2.4 | |
| 5 | (g) | Data point (3.99, 11.42) plotted correctly |
| [1] | 1.1 | |
| 5 | (h) | The fit would be worse. |
| Answer | Marks |
|---|---|
| point which is perhaps an outlier. | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 2.2b | |
| 2.4 | may refer to residuals or correlation | |
| Rank BCG | 1 | 3 |
| Rank Measles | 1 | 3 |
Question 5:
5 | (a) | Because (the grouping of points on) the scatter diagram
does not appear to be elliptical
due to the outlier/anomalous point | E1
E1
[2] | 3.5a
2.4 | Or: not bivariate Normal
5 | (b) | Rank BCG 1 3 6 2 7 5 4 8
Rank Measles 1 3 4 2 5 7 6 8
Evaluates soi
Spearman’s ra2nk coeff = (= 0.8095… or 0.81)
∑𝑑𝑑 (= 16)
17 | M1
M1
A1
[3] | 1.1
1.1
1.1 | For ranking BCG & Measles
(may be reversed or implied from d
values)
BC | Use of percentage values
is M0 M0
5 | (c) | 21
H : There is no association between (vaccination rates
0
for) BCG and measles in the population
H : There is some association between (vaccination
1
rates for) BCG and measles in the population
For n = 8, 5% critical value is (±)0.7381
0.8095 > 0.7381 Reject H oe
0
There is sufficient evidence to suggest that there is
some association between (vaccination rates for) BCG
and measles (in the population) | B1
B1
B1
M1
A1
[5] | 3.3
1.2
3.4
1.1
2.2b | Need to see population in one or
other of the hypotheses
For comparison with sensible critical
value, |r| < 1 and conclusion correct
s
FT their r
s
Not too assertive and includes context | SC if outlier excluded in
(b) may still score 5/5
For n = 7, c.v. is 0.7857
0.7143 < 0.7857
Do not reject H
0
Insufficient evidence…
5 | (d) | y = -4.80x + 24.49 awrt
x = doctors, y = unemployment | B1
B1
[2] | 1.1
1.1 | BC
Variables defined (other letters may
be used)
5 | (e) | 14.89 [14.8, 15.0] | B1
[1] | 1.1
5 | (f) | Not reliable since extrapolation oe | E1
[1] | 2.4
5 | (g) | Data point (3.99, 11.42) plotted correctly | B1
[1] | 1.1
5 | (h) | The fit would be worse.
The regression line might no longer be valid due to this
point which is perhaps an outlier. | E1
E1
[2] | 2.2b
2.4 | may refer to residuals or correlation
Rank BCG | 1 | 3 | 6 | 2 | 7 | 5 | 4 | 8
Rank Measles | 1 | 3 | 4 | 2 | 5 | 7 | 6 | 85 A student is investigating immunisation. He wonders if there is any relationship between the percentage of young children who have been given measles vaccine and the percentage who have been given BCG vaccine in various countries.
He takes a random sample of 8 countries and finds the data for the two variables. The spreadsheet in Fig. 5.1 shows the values obtained, together with a scatter diagram which illustrates the data.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-5_910_1653_541_246}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item The student decides that a test based on Pearson's product moment correlation coefficient is not valid. Explain why he comes to this conclusion.
The student carries out a test based on Spearman's rank correlation coefficient.
\item Calculate the value of Spearman's rank correlation coefficient.
\item Carry out a test based on this coefficient at the $5 \%$ significance level to investigate whether there is any association between measles and BCG vaccination levels.
The student then decides to investigate the relationship between number of doctors per 1000 people in a country and unemployment rate in that country (unemployment rate is the percentage of the working age population who are not in paid work). He selects a random sample of 6 countries. The spreadsheet in Fig. 5.2 shows the values obtained, together with a scatter diagram which illustrates the data.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{882f9f3c-40d8-4abb-822a-49bd505a33ea-6_776_1649_495_248}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
\end{center}
\end{figure}
\item Use your calculator to write down the equation of the regression line of unemployment rate on doctors per 1000.
\item Use the regression line to estimate the unemployment rate for a country with 2.00 doctors per 1000.
\item Comment briefly on the reliability of your answer to part (e).
The student decides to add the data for another country with 3.99 doctors per 1000 and unemployment rate 11.42 to his diagram.
\item Add this point to the scatter diagram in the Printed Answer Booklet.
\item Without doing any further calculations, comment on what difference, if any, including this extra data point would make to the usefulness of a regression line of unemployment rate on doctors per 1000.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Minor 2020 Q5 [17]}}