| Exam Board | OCR MEI |
|---|---|
| Module | Further Statistics Major (Further Statistics Major) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Two-tailed test for any correlation |
| Difficulty | Standard +0.3 This is a straightforward application of hypothesis testing for correlation with standard procedures: (a) requires a routine t-test calculation with n=60 and r=0.3231, comparing to critical values at 10% significance; (b) asks for a standard validity condition (bivariate normality). The table interpretation adds minimal complexity. Slightly above average difficulty due to being Further Maths content, but mechanically routine. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| Correlations | PM10 | SPEED | \(\mathrm { NO } _ { 2 }\) | \(\mathrm { O } _ { 3 }\) | PM25 | \(\mathrm { SO } _ { 2 }\) |
| PM10 | 1.00 | |||||
| SPEED | 0.08* | 1.00 | ||||
| \(\mathrm { NO } _ { 2 }\) | 0.59* | 0.25* | 1.00 | |||
| \(\mathbf { O } _ { \mathbf { 3 } }\) | -0.05* | -0.04* | -0.30* | 1.00 | ||
| PM25 | 0.85* | -0.01 | 0.56* | -0.02 | 1.00 | |
| \(\mathrm { SO } _ { 2 }\) | 0.42* | 0.15* | 0.73* | -0.63* | 0.40* | 1.00 |
| Effect size | ||
| 0.1 | Small | ||
| 0.3 | Medium | ||
| 0.5 | Large |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | H : ρ = 0 , H : ρ ≠ 0 (two-tailed test) |
| Answer | Marks |
|---|---|
| nitrogen dioxide level | B1 |
| Answer | Marks |
|---|---|
| [5] | 3.3 |
| Answer | Marks |
|---|---|
| 2.2b | For both hypotheses |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (b) | The population from which the data are drawn must be |
| bivariate Normally distributed. | B1 | |
| [1] | 2.5 | Do not allow ‘Data must be bivariate Normally distributed’ but |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (c) | For very large samples, the sample correlation |
| Answer | Marks |
|---|---|
| which are very close to zero are of very little real use. | E1 |
| Answer | Marks |
|---|---|
| [2] | 3.5a |
| 3.5a | oe eg ‘large sample size means lower values of r can be |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (d) | Although four out of five of the correlation coefficients |
| Answer | Marks |
|---|---|
| 2 | E1 |
| Answer | Marks |
|---|---|
| [2] | 2.2b |
Question 6:
6 | (a) | H : ρ = 0 , H : ρ ≠ 0 (two-tailed test)
0 1
where ρ is the population correlation coefficient
(between wind-speed and nitrogen dioxide level)
For n = 10, 5% critical value = 0.2144
Since 0.3231 > 0.2144 the result is significant. There is
sufficient evidence to reject H
0
There is sufficient evidence at the 10% level to suggest
that there is correlation between wind-speed and
nitrogen dioxide level | B1
B1
B1
M1
A1
[5] | 3.3
2.5
3.4
1.1
2.2b | For both hypotheses
For defining ρ
For critical value
For comparison leading to a
conclusion
FT for conclusion in words
6 | (b) | The population from which the data are drawn must be
bivariate Normally distributed. | B1
[1] | 2.5 | Do not allow ‘Data must be bivariate Normally distributed’ but
condone ‘Data must be from a bivariate Normal distribution’
6 | (c) | For very large samples, the sample correlation
coefficient is very often significant.
However as in this case those significant correlations
which are very close to zero are of very little real use. | E1
E1
[2] | 3.5a
3.5a | oe eg ‘large sample size means lower values of r can be
significant’
6 | (d) | Although four out of five of the correlation coefficients
are statistically significant, very little use can be made
of most of them as the effect sizes are small.
However the effect size of 0.25 between wind-speed
and NO suggests that the windier it is, the higher the
2
level of NO pollution.
2 | E1
E1
[2] | 2.2b
2.2b6 A pollution control officer is investigating a possible link between the levels of various pollutants in the air and the speed of the wind at various sites. A random sample of 60 values of the windspeed together with the levels of a variety of pollutants is taken at a particular site. The product moment correlation coefficient between wind-speed and nitrogen dioxide level is 0.3231 .
\begin{enumerate}[label=(\alph*)]
\item Carry out a hypothesis test at the $10 \%$ significance level to investigate whether there is any correlation between wind-speed and nitrogen dioxide level.
\item State the condition required for the test carried out in part (a) to be valid.
Table 6.1 shows the values of the product moment correlation coefficient between 5 different measures of pollution and also wind-speed for a very large random sample of values at another site. Those correlations that are significant at the $10 \%$ level are denoted by a * after the value of the correlation.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Correlations & PM10 & SPEED & $\mathrm { NO } _ { 2 }$ & $\mathrm { O } _ { 3 }$ & PM25 & $\mathrm { SO } _ { 2 }$ \\
\hline
PM10 & 1.00 & & & & & \\
\hline
SPEED & 0.08* & 1.00 & & & & \\
\hline
$\mathrm { NO } _ { 2 }$ & 0.59* & 0.25* & 1.00 & & & \\
\hline
$\mathbf { O } _ { \mathbf { 3 } }$ & -0.05* & -0.04* & -0.30* & 1.00 & & \\
\hline
PM25 & 0.85* & -0.01 & 0.56* & -0.02 & 1.00 & \\
\hline
$\mathrm { SO } _ { 2 }$ & 0.42* & 0.15* & 0.73* & -0.63* & 0.40* & 1.00 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Table 6.1}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Table 6.2 shows standard guidelines for effect sizes.}
\begin{tabular}{ | c | c | }
\hline
\begin{tabular}{ c }
Product moment \\
correlation coefficient \\
\end{tabular} & Effect size \\
\hline
0.1 & Small \\
\hline
0.3 & Medium \\
\hline
0.5 & Large \\
\hline
\end{tabular}
\end{center}
\end{table}
Table 6.2
The officer analyses these data for effect size.
\item Explain how the very large sample size relates to the interpretation of the correlation coefficients shown in Table 6.1.
\item Comment briefly on what the pollution control officer might conclude from these tables, relevant to her investigation into wind-speed and pollutant levels.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Statistics Major 2020 Q6 [10]}}