| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2018 |
| Session | September |
| Marks | 8 |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | One-tailed test for positive correlation |
| Difficulty | Moderate -0.3 This is a straightforward hypothesis test for correlation requiring students to look up a critical value in a provided table (n=7, 5% one-tail gives 0.6694), compare it to the calculated r=0.894, and state standard hypotheses and conclusion. It's slightly easier than average because the table is given, the comparison is simple, and it's a routine application of a standard procedure with no problem-solving required. |
| Spec | 2.05a Hypothesis testing language: null, alternative, p-value, significance2.05g Hypothesis test using Pearson's r |
| 1-tail test 2-tail test | 5\% | 2.5\% | 1\% | 0.5\% |
| 10\% | 5\% | 2\% | 1\% | |
| \(n\) | ||||
| 1 | - | - | - | - |
| 2 | - | - | - | - |
| 3 | 0.9877 | 0.9969 | 0.9995 | 0.9999 |
| 4 | 0.9000 | 0.9500 | 0.9800 | 0.9900 |
| 5 | 0.8054 | 0.8783 | 0.9343 | 0.9587 |
| 6 | 0.7293 | 0.8114 | 0.8822 | 0.9587 |
| 7 | 0.6694 | 0.7545 | 0.8329 | 0.9745 |
| 8 | 0.6215 | 0.7067 | 0.7887 | 0.8343 |
| 9 | 0.5882 | 0.6664 | 0.7498 | 0.7977 |
| 10 | 0.5494 | 0.6319 | 0.7155 | 0.7646 |
11 In an experiment involving a bivariate distribution ( $X , Y$ ) a random sample of 7 pairs of values was obtained and Pearson's product-moment correlation coefficient $r$ was calculated for these values.\\
(i) The value of $r$ was found to be 0.894 . Use the table below to test, at the $5 \%$ significance level, whether there is positive linear correlation in the population, stating your hypotheses and conclusion clearly.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
1-tail test 2-tail test & 5\% & 2.5\% & 1\% & 0.5\% \\
\hline
& 10\% & 5\% & 2\% & 1\% \\
\hline
$n$ & & & & \\
\hline
1 & - & - & - & - \\
\hline
2 & - & - & - & - \\
\hline
3 & 0.9877 & 0.9969 & 0.9995 & 0.9999 \\
\hline
4 & 0.9000 & 0.9500 & 0.9800 & 0.9900 \\
\hline
5 & 0.8054 & 0.8783 & 0.9343 & 0.9587 \\
\hline
6 & 0.7293 & 0.8114 & 0.8822 & 0.9587 \\
\hline
7 & 0.6694 & 0.7545 & 0.8329 & 0.9745 \\
\hline
8 & 0.6215 & 0.7067 & 0.7887 & 0.8343 \\
\hline
9 & 0.5882 & 0.6664 & 0.7498 & 0.7977 \\
\hline
10 & 0.5494 & 0.6319 & 0.7155 & 0.7646 \\
\hline
\end{tabular}
\end{center}
Scatter diagrams for four sets of bivariate data, are shown.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_380_371_301_191}
\captionsetup{labelformat=empty}
\caption{Diagram A}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_373_373_301_628}
\captionsetup{labelformat=empty}
\caption{Diagram B}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1064}
\captionsetup{labelformat=empty}
\caption{Diagram C}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{85de9a39-f8be-40ee-b0c8-e2e632be93d8-8_378_373_301_1503}
\captionsetup{labelformat=empty}
\caption{Diagram D}
\end{center}
\end{figure}
It is given that $r = 0.894$ for one of these diagrams.\\
(ii) For each of the other diagrams, state how you can tell that $r \neq 0.894$.
\hfill \mbox{\textit{OCR H240/02 2018 Q11 [8]}}