14.4 Interpreting correlation coefficients — Practise with help

1. The correlation is positive and strong, because the points rise and lie close to a line.
2. The correlation is negative and strong, because the sign is negative and $-0.8$ is close to $-1$.
3. Each tail holds $2.5\%$, because the 5% is shared equally.

If question 3 was hard, revise Section 14.1 before you start.

A p-value is compared with the significance level, not with $r$.

Compare $0.017$ with $0.05$. If the p-value is smaller, the test rejects $H_0$.

Since

the test rejects $H_0$. There is sufficient evidence at the 5% level of a positive linear correlation between the two variables. The phrase "positive linear" keeps the conclusion to what the test measured.

A two-tailed test compares $|r|$ with the critical value.

Compare $0.35$ with $0.361$. Decide whether the coefficient reaches the critical value.

The coefficient is below the critical value, so the test retains $H_0$. There is not sufficient evidence at the 5% level of a linear correlation.

This is close to the boundary. A small change in the data, or a slightly larger sample, could move the coefficient past the critical value.

The two tests use different critical values. The tail must be chosen from the question, before the data are used.

For part (a), compare $0.44$ with $0.378$. For part (b), compare $0.44$ with $0.444$. Notice how close part (b) is to the boundary.

(a) For the one-tailed test, $0.44 > 0.378$, so the test rejects $H_0$. There is sufficient evidence at the 5% level of a positive linear correlation.

(b) For the two-tailed test, $0.44 < 0.444$, so the test retains $H_0$. There is not sufficient evidence at the 5% level of a linear correlation.

The same value of $r$ gives two decisions. The two-tailed critical value is larger, because the 5% is shared between two tails, so each tail is harder to reach.

The alternative is negative, so the test rejects only when $r$ is below the negative of the critical value.

Compare $-0.52$ with $-0.337$. Decide which value is further from zero in the negative direction.

The coefficient is below the critical value, so the test rejects $H_0$. There is sufficient evidence at the 5% level of a negative linear correlation between the age of the machine and its efficiency. Older machines in this sample tend to be less efficient.

The conclusion is about an association, not a cause. A machine may lose efficiency for a reason that also grows with its age, such as the number of hours it has run.

The coefficient is the same in both samples. Only the critical value changes.

Compare $0.35$ with each critical value in turn.

For 30 pairs,

so the test retains $H_0$.

For 60 pairs,

so the test rejects $H_0$.

The larger sample gives a smaller critical value, so a moderate coefficient can be significant when it is based on more pairs. A correlation must always be reported with the number of pairs used.

A supported conclusion uses words such as "associated with". An unsupported one uses the word "causes".

Write the association first, then write the causal claim that the test cannot support.

A supported conclusion: "There is evidence of a positive association between hours of sleep and test score in this population."

Too far: "Sleeping more causes a higher test score." The correlation shows that the two variables move together. A third factor, such as time available to revise, could drive both.