| Exam Board | Edexcel |
|---|---|
| Module | Paper 3 (Paper 3) |
| Year | 2020 |
| Session | October |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Describe correlation from scatter diagram |
| Difficulty | Moderate -0.8 This is a straightforward hypothesis testing question requiring standard procedures: describing correlation from a scatter diagram, stating hypotheses for PMCC test, comparing to critical values from tables, and making a basic interpretation. All parts are routine recall and application of standard techniques with no problem-solving insight required. The multi-part structure and context from the large data set add length but not conceptual difficulty. |
| Spec | 2.01d Select/critique sampling: in context2.02d Informal interpretation of correlation2.05g Hypothesis test using Pearson's r |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | Negative | B1 |
| (b)(i) | Rainfall (mm) or Pressure (hPa or Pascals or hectopascals or mb or millibars) | B1; B1ft |
| (b)(ii) | ||
| (c) | \(H_0 : \rho = 0\); \(H_1 : \rho \neq 0\). Critical value: \(-0.361(0)\). \(r < -0.3610\) so significant result and there is evidence of a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity. | B1; M1; A1 |
| (d) | Humidity is high and there is evidence of correlation and \(r < 0\). So expect amount of sunshine to be lower than the average for Heathrow(oe). | B1 |
| (a) | Negative | B1 | For stating negative. "Negative skew" is B0 though. |
|---|---|---|---|
| (b)(i) | Rainfall (mm) or Pressure (hPa or Pascals or hectopascals or mb or millibars) | B1; B1ft | B1 for mentioning "rainfall" (allow "rain" or "precipitation") or "pressure" (if more than 1 answer both must be correct). NB the other quantitative variable for Perth is: Daily Mean Wind Speed and scores B0. [Not allowed "wind speed" since $r = +0.15$ and in winter might expect wind to raise temp]. B1ft for giving the correct units. If Daily Mean Wind Speed (kn) or knots "Wind speed" and "knots" would score B0B0 [Not allowed "wind speed" since $r = +0.15$ and in winter might expect wind to raise temp]. |
| (b)(ii) | | | |
| (c) | $H_0 : \rho = 0$; $H_1 : \rho \neq 0$. Critical value: $-0.361(0)$. $r < -0.3610$ so significant result and there is evidence of a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity. | B1; M1; A1 | B1 for both hypotheses correct in terms of $\rho$. M1 for the correct critical value compatible with their $H_1$; allow $\pm 0.361(0)$. If the hypotheses are 1-tail then allow cv of $\pm 0.3061$. e.g. Alternative hypothesis with $r < \pm 0.377$ implies a one-tail test or $H_0$ and $H_1$ in words saying "$H_0$: there is no correlation, $H_1$: there is correlation" is two-tail. If there are no hypotheses (or they are nonsensical) assume 2-tail so M1 for $\pm 0.361(0)$. A1 for a correct conclusion in context based on comparing $-0.377$ with their cv. Condone incorrect inequality e.g. $-0.3610 < -0.377$ as long as they reject $H_0$. Do not accept contradictory statements such as "accept $H_0$ so there is evidence of ...". Can say "support for Stav's belief" (o.e.e.g. "claim") or "evidence of a correlation between sunshine and humidity" condone "negative correlation" or comments such as "if humidity is high amount of sunshine will be low". |
| (d) | Humidity is high and there is evidence of correlation and $r < 0$. So expect amount of sunshine to be lower than the average for Heathrow(oe). | B1 | B1 for stating low amount of sunshine (o. e.) and some reference to $r < 0$ or fog. Check for the following 2 features: (i) low sunshine: allow $\leq 5$ hrs (LDS mean for 2015 is 5.3, humidity 97% is 4.1, $\geq 97\%$ is 3.1); (ii) negative correlation may be described in words e.g. "high humidity gives low sunshine" or fog (LDS says >95% humidity is foggy) so less sunshine. |\begin{enumerate}
\item A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.
\end{enumerate}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2b63aa7f-bc50-4422-8dc0-e661b521c221-04_722_709_376_677}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
(a) Describe the correlation.
The variable on the $x$-axis is Daily Mean Temperature measured in ${ } ^ { \circ } \mathrm { C }$.\\
(b) Using your knowledge of the large data set,\\
(i) suggest which variable is on the $y$-axis,\\
(ii) state the units that are used in the large data set for this variable.
Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow.
He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains $r = - 0.377$\\
(c) Carry out a suitable test to investigate Stav's belief at a $5 \%$ level of significance. State clearly
\begin{itemize}
\item your hypotheses
\item your critical value
\end{itemize}
On a random day at Heathrow the Daily Maximum Relative Humidity was 97\%\\
(d) Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.
\hfill \mbox{\textit{Edexcel Paper 3 2020 Q2 [7]}}