14.4 Interpreting correlation coefficients — Practise

Work through these on your own. Decide the tail before you compare a p-value or a coefficient.

The questions and the answers below come from a set that is checked by computer every time the book is built.

A study of 15 pairs gives a correlation coefficient $r = 0.62$ . The test is one-tailed with $H_1: \rho > 0$ at the 5% level, and the p-value is $0.007$ . State, with a symbol, how the p-value compares with the significance level, and name the decision.
A sample of 20 pairs gives $r = 0.22$ and a two-tailed p-value of $0.35$ . State, with a symbol, how the p-value compares with the 5% significance level, and name the decision.
A sample of 30 pairs gives $r = 0.42$ and a two-tailed p-value of $0.021$ . State, with a symbol, how the p-value compares with the 1% significance level, and name the decision.
A sample of 30 pairs gives $r = 0.42$ . The test is two-tailed at the 5% level and the critical value for 30 pairs is $0.361$ . State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
A sample of 30 pairs gives $r = 0.35$ . The test is two-tailed at the 5% level and the critical value for 30 pairs is $0.361$ . State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
A sample of 20 pairs gives $r = 0.44$ . The test is one-tailed with $H_1: \rho > 0$ at the 5% level and the critical value is $0.378$ . State, with a symbol, how $r$ compares with the critical value, and name the decision.
The same 20 pairs give $r = 0.44$ . The test is now two-tailed at the 5% level and the critical value is $0.444$ . State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
A sample of 25 pairs gives $r = -0.52$ . The test is one-tailed with $H_1: \rho < 0$ at the 5% level and the critical value is $0.337$ . State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
A sample of 25 pairs gives $r = -0.28$ . The test is one-tailed with $H_1: \rho < 0$ at the 5% level and the critical value is $0.337$ . State, with a symbol, how $|r|$ compares with the critical value, and name the decision.
The two-tailed 5% critical value for a correlation is $0.361$ for 30 pairs and $0.254$ for 60 pairs. State, with a symbol, how the critical value for 60 pairs compares with the critical value for 30 pairs, and explain what this means for a fixed value of $r$ .
A correlation coefficient is $r = -0.78$ . Write down $|r|$ , the value a two-tailed test compares with the critical value.
A two-tailed test of a correlation uses the 5% significance level. Find the probability placed in each tail.

Answers and concise solutions

1. The comparison and the decision

Answer: $0.007 < 0.05$ , so the test rejects $H_0$ . There is sufficient evidence at the 5% level of a positive linear correlation.

2. The comparison and the decision

Answer: $0.35 > 0.05$ , so the test retains $H_0$ . There is not sufficient evidence at the 5% level of a linear correlation.

3. The comparison and the decision

Answer: $0.021 > 0.01$ , so the test retains $H_0$ at the 1% level, although it rejects at the 5% level. The level must be reported with the decision.

4. The comparison and the decision

Answer: $0.42 > 0.361$ , so the test rejects $H_0$ . There is sufficient evidence at the 5% level of a linear correlation.

5. The comparison and the decision

Answer: $0.35 < 0.361$ , so the test retains $H_0$ . There is not sufficient evidence at the 5% level of a linear correlation.

6. The comparison and the decision

Answer: $0.44 > 0.378$ , so the test rejects $H_0$ . There is sufficient evidence at the 5% level of a positive linear correlation.

7. The comparison and the decision

Answer: $0.44 < 0.444$ , so the test retains $H_0$ . The same coefficient that rejected a one-tailed test does not reach the larger two-tailed critical value.

8. The comparison and the decision

Answer: $|-0.52| = 0.52 > 0.337$ , so the test rejects $H_0$ . There is sufficient evidence at the 5% level of a negative linear correlation.

9. The comparison and the decision

Answer: $|-0.28| = 0.28 < 0.337$ , so the test retains $H_0$ . There is not sufficient evidence at the 5% level of a negative linear correlation.

10. The comparison

Answer: $0.254 < 0.361$ . A larger sample gives a smaller critical value, so a moderate correlation is more likely to be significant when it is based on more pairs.

11. The value of $|r|$

Answer: $|-0.78| = 0.78$ . A two-tailed test uses the size of the coefficient and ignores the sign.

12. The probability in each tail

Answer: The 5% is shared equally, so each tail holds $\dfrac{0.05}{2} = 0.025$ .