Q1
Answer
The following table shows the critical values for the product moment correlation coefficient at the specified significance levels, for a one tailed test, for a sample of size $n$.
$n$$0.1$$0.05$$0.025$$0.01$
$4$$0.8000$$0.9000$$0.9500$$0.9800$
$5$$0.6870$$0.8054$$0.8783$$0.9343$
$6$$0.6084$$0.7293$$0.8114$$0.8822$
$7$$0.5509$$0.6694$$0.7545$$0.8329$

Use this to test whether six data points with a PMCC value of $0.7$ are positively correlated at the 5% significance level.
$\mathrm{H}_0: \rho = 0$ and $\mathrm{H}_1: \rho > 0$ <br> $|r| < 0.7293$, reject $\mathrm{H}_1$. There is insufficient evidence to suggest the two variables are positively correlated.
Q2
Answer
The following table shows the critical values for the product moment correlation coefficient at the specified significance levels, for a one tailed test, for a sample of size $n$.
$n$$0.1$$0.05$$0.025$$0.01$
$4$$0.8000$$0.9000$$0.9500$$0.9800$
$5$$0.6870$$0.8054$$0.8783$$0.9343$
$6$$0.6084$$0.7293$$0.8114$$0.8822$
$7$$0.5509$$0.6694$$0.7545$$0.8329$

Use this to test whether four data points with a PMCC value of $-0.97$ are negatively correlated at the 2.5% significance level.
$\mathrm{H}_0: \rho = 0$ and $\mathrm{H}_1: \rho < 0$ <br> $|r| > 0.9500$, reject $\mathrm{H}_0$. There is evidence to suggest the two variables are negatively correlated.
Q3
Answer
The following table shows the critical values for the product moment correlation coefficient at the specified significance levels, for a one tailed test, for a sample of size $n$.
$n$$0.1$$0.05$$0.025$$0.01$
$4$$0.8000$$0.9000$$0.9500$$0.9800$
$5$$0.6870$$0.8054$$0.8783$$0.9343$
$6$$0.6084$$0.7293$$0.8114$$0.8822$
$7$$0.5509$$0.6694$$0.7545$$0.8329$

Use this to test whether seven data points with a PMCC value of $-0.7$ are correlated at the 5% significance level.
$\mathrm{H}_0: \rho = 0$ and $\mathrm{H}_1: \rho \neq 0$ <br> $|r| < 0.7545$ (two tailed test, halve the significance level compared to one tailed), reject $\mathrm{H}_1$. There is insufficient evidence to suggest the two variables are correlated.
Q4
Answer
The following table shows the critical values for the product moment correlation coefficient at the specified significance levels, for a one tailed test, for a sample of size $n$.
$n$$0.1$$0.05$$0.025$$0.01$
$4$$0.8000$$0.9000$$0.9500$$0.9800$
$5$$0.6870$$0.8054$$0.8783$$0.9343$
$6$$0.6084$$0.7293$$0.8114$$0.8822$
$7$$0.5509$$0.6694$$0.7545$$0.8329$
Over a seven year span between 2001 and 2007, the PMCC between the number of people who drowned in a pool and the number of films Nicholas Cage appeared in that year is $0.8213$.
  1. Test, at the 2.5% level of significance, whether these are positively correlated.
  2. The number of films Nicholas Cage appeared in in 2008 doubled compared to 2007. A student sugggests that the number of deaths due to drowning in pools also doubled in 2008. Explain whether the student is correct.
  1. $\mathrm{H}_0: \rho = 0$ and $\mathrm{H}_1: \rho > 0$ <br> $|r| > 0.7545$, reject $\mathrm{H}_0$. There is sufficient evidence to suggest that these are positively correlated.
  2. It is likely that the correlation is due to coincidence rather than one causing the other, so the student is incorrect to make the statement.
Q5
Answer
The following table shows the critical values for the product moment correlation coefficient at the specified significance levels, for a one tailed test, for a sample of size $n$.
$n$$0.1$$0.05$$0.025$$0.01$
$4$$0.8000$$0.9000$$0.9500$$0.9800$
$5$$0.6870$$0.8054$$0.8783$$0.9343$
$6$$0.6084$$0.7293$$0.8114$$0.8822$
$7$$0.5509$$0.6694$$0.7545$$0.8329$
Over a six week period, the PMCC for the number of ice creams sold at a supermarket and the average temperature that week was found to be $0.8$.
  1. Suggest a reason for why these could be correlated.
  2. Test, at the 5% level of significance, whether these are correlated.
  1. People are more likely to buy ice cream when the weather is nice.
  2. $\mathrm{H}_0: \rho = 0$ and $\mathrm{H}_1: \rho \neq 0$ <br> $|r| < 0.8114$ (two tailed), reject $\mathrm{H}_1$. There is insufficient evidence to suggest that these are correlated.
Q6
Answer
The following table shows the critical values for the product moment correlation coefficient at the specified significance levels, for a one tailed test, for a sample of size $n$.
$n$$0.1$$0.05$$0.025$$0.01$
$4$$0.8000$$0.9000$$0.9500$$0.9800$
$5$$0.6870$$0.8054$$0.8783$$0.9343$
$6$$0.6084$$0.7293$$0.8114$$0.8822$
$7$$0.5509$$0.6694$$0.7545$$0.8329$
The PMCC for the number of pupils who were late to school and the number of detentions a school handed out to their pupils each day over a week was found to be $0.85$.
  1. Suggest a reason for why these could be correlated.
  2. Test, at the 5% level of significance, whether these are correlated.
  1. Pupils might get detentions for being late.
  2. $\mathrm{H}_0: \rho = 0$ and $\mathrm{H}_1: \rho \neq 0$ <br> $|r| < 0.8783$ (two tailed), reject $\mathrm{H}_1$. There is insufficient evidence to suggest that these are correlated.