| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Topic | Hypothesis test of Pearson’s product-moment correlation coefficient |
| Type | Use critical value table directly |
| Difficulty | Moderate -0.8 This is a straightforward application of hypothesis testing for correlation requiring only: (1) justifying a one-tail test with basic contextual reasoning (longer gestation → heavier babies), and (2) comparing a given r-value (0.722) to a critical value from a provided table (0.6851). No calculations needed, just table lookup and comparison—simpler than typical hypothesis test questions that require computing test statistics. |
| Spec | 5.08a Pearson correlation: calculate pmcc5.08d Hypothesis test: Pearson correlation |
| \multirow{2}{*}{} | 1-tail test | 5\% | 2.5\% | 1\% | 0.5\% |
| 2-tail test | 10\% | 5\% | 2.5\% | 1\% | |
| \multirow{4}{*}{\(n\)} | 10 | 0.5494 | 0.6319 | 0.7155 | 0.7646 |
| 11 | 0.5214 | 0.6021 | 0.6851 | 0.7348 | |
| 12 | 0.4973 | 0.5760 | 0.6581 | 0.7079 | |
| 13 | 0.4762 | 0.5529 | 0.6339 | 0.6835 |
| Answer | Marks | Guidance |
|---|---|---|
| Very likely weight will increase with time; or He is only looking for positive correlation | B1 [1] | Or e.g. "Expect weight to increase with time"; "Foetuses grow". Ignore all else |
| Answer | Marks | Guidance |
|---|---|---|
| \(H_0: \rho = 0\); \(H_1: \rho > 0\) where \(\rho\) is the correlation coefficient for the population | B1, B1 | B1B0 for 1 error e.g. undefined \(\rho\) or 2-tail. For hypotheses in words: \(H_0\): no correlation between time and weight; \(H_1\): positive correlation — omission of "positive": B0B0 |
| Compare 0.722 with 0.6851; Reject \(H_0\) | M1, M1 | May be implied by conclusion |
| There is evidence of (positive linear) correlation between time from conception to birth and weight of new-born babies | A1 [5] | In context, not definite. Allow without "positive" and without "linear" |
## Question 10(a):
| Very likely weight will increase with time; or He is only looking for positive correlation | B1 [1] | Or e.g. "Expect weight to increase with time"; "Foetuses grow". Ignore all else |
|---|---|---|
---
## Question 10(b):
| $H_0: \rho = 0$; $H_1: \rho > 0$ where $\rho$ is the correlation coefficient for the population | B1, B1 | B1B0 for 1 error e.g. undefined $\rho$ or 2-tail. For hypotheses in words: $H_0$: no correlation between time and weight; $H_1$: positive correlation — omission of "positive": B0B0 |
|---|---|---|
| Compare 0.722 with 0.6851; Reject $H_0$ | M1, M1 | May be implied by conclusion |
| There is evidence of (positive linear) correlation between time from conception to birth and weight of new-born babies | A1 [5] | In context, not definite. Allow without "positive" and without "linear" |
---10 A researcher plans to carry out a statistical investigation to test whether there is linear correlation between the time ( $T$ weeks) from conception to birth, and the birth weight ( $W$ grams) of new-born babies.
\begin{enumerate}[label=(\alph*)]
\item Explain why a 1-tail test is appropriate in this context.
The researcher records the values of $T$ and $W$ for a random sample of 11 babies. They calculate Pearson's product-moment correlation coefficient for the sample and find that the value is 0.722 .
\item Use the table below to carry out the test at the $1 \%$ significance level.
\section*{Critical values of Pearson's product-moment correlation coefficient.}
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
\multirow{2}{*}{} & 1-tail test & 5\% & 2.5\% & 1\% & 0.5\% \\
\hline
& 2-tail test & 10\% & 5\% & 2.5\% & 1\% \\
\hline
\multirow{4}{*}{$n$} & 10 & 0.5494 & 0.6319 & 0.7155 & 0.7646 \\
\hline
& 11 & 0.5214 & 0.6021 & 0.6851 & 0.7348 \\
\hline
& 12 & 0.4973 & 0.5760 & 0.6581 & 0.7079 \\
\hline
& 13 & 0.4762 & 0.5529 & 0.6339 & 0.6835 \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2021 Q10 [6]}}