11.4 Testing the Significance of the Correlation Coefficient

Step 1 — Identify the population coefficient. The population correlation coefficient is denoted \(\rho\) (the Greek letter "rho"), and it is unknown because we almost never have data for the entire population.

Step 2 — Identify the sample coefficient. The sample correlation coefficient is \(r\), computed directly from the sample data, so it is known.

Step 3 — State the relationship between them. Because \(\rho\) is unknown, \(r\) serves as our estimate of it, and the hypothesis test uses \(r\) together with the sample size \(n\) to decide whether \(\rho\) is significantly different from zero.

Answer: \(\rho\) is the population correlation coefficient (unknown); \(r\) is the sample correlation coefficient (known), and \(r\) estimates \(\rho\).

Step 1 — Write the hypotheses in symbols.

Step 2 — State \(H_{0}\) in words. The population correlation coefficient IS NOT significantly different from zero; there IS NOT a significant linear relationship (correlation) between \(x\) and \(y\) in the population.

Step 3 — State \(H_{a}\) in words. The population correlation coefficient IS significantly DIFFERENT FROM zero; there IS a significant linear relationship (correlation) between \(x\) and \(y\) in the population.

Step 4 — Note what the hypotheses concern. Both statements are about \(\rho\), the population parameter — not about \(r\), which is already known exactly from the sample.

Answer: \(H_{0}: \rho = 0\) (no significant linear relationship in the population); \(H_{a}: \rho eq 0\) (there is a significant linear relationship in the population).

Step 1 — Recall what \(r\) measures. It reports the strength and direction of the linear association in the sample that was collected — nothing more.

Step 2 — Consider small samples. With few observations, chance alone can produce a fairly high correlation even when \(x\) and \(y\) are unrelated in the population; there are too few points to constrain the pattern.

Step 3 — Consider large samples. With many observations, accidental patterns tend to wash out, so the same value of \(r\) is far less likely to be a fluke.

Step 4 — Draw the conclusion. Two studies reporting identical values of \(r\) can differ greatly in how convincing they are, which is why the test always uses \(r\) and \(n\) together, never \(r\) alone.

Answer: The same \(r\) is more convincing from a larger sample, because a small sample can produce a high correlation by chance while a large one rarely can.

Step 1 — Write the formula for the test statistic.

Step 2 — State the degrees of freedom. The test statistic follows a t-distribution with

degrees of freedom, where \(n\) is the number of observed data points.

Step 3 — Note two extra facts worth remembering. The statistic \(t\) has the same sign as \(r\), and the p-value is the combined area in both tails of the t-distribution (because \(H_a\) is two-sided).

Answer: \(t = \dfrac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}\), with \(df = n - 2\).

Step 1 — Compare the p-value with \(\alpha\). The p-value is 0.018 and the significance level is \(\alpha = 0.05\). Since \(0.018 < 0.05\), the p-value is less than \(\alpha\).

Step 2 — Make the decision. When the p-value < \(\alpha\), we reject the null hypothesis \(H_{0}: \rho = 0\).

Step 3 — Write the conclusion. There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero.

Step 4 — Note what this licenses. If the scatter plot also shows a linear trend, the regression line may be used to model the relationship and predict \(y\) within the domain of observed \(x\) values.

Answer: Reject \(H_{0}\): there is sufficient evidence that the correlation is significantly different from zero, so a significant linear relationship exists.

Step 1 — Compare the p-value with \(\alpha\). The p-value is 0.214 and \(\alpha = 0.05\). Since \(0.214 > 0.05\), the p-value is NOT less than \(\alpha\).

Step 2 — Make the decision. Do NOT reject the null hypothesis \(H_{0}: \rho = 0\).

Step 3 — Interpret the result. There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\), because the correlation coefficient is not significantly different from zero.

Step 4 — Say what this means for the line. Since \(r\) is not significant, the regression line should NOT be used to model the relationship or to predict \(y\) values.

Answer: Do not reject \(H_{0}\); the correlation is not significant, so the regression line cannot be used for prediction.

Step 1 — Find the degrees of freedom. \(df = n - 2 = 22 - 2 = 20\).

Step 2 — Set up the critical values. With critical value 0.423, the band is from \(-0.423\) to \(+0.423\).

Step 3 — Apply the rule. \(r\) is significant if it lies outside the band. Here \(r = 0.44 > 0.423\), so \(r\) falls outside the interval.

Step 4 — Conclude. \(r\) is significant, and the line may be used for prediction (assuming the scatter plot shows a linear trend).

Answer: Yes — since \(0.44 > 0.423\), \(r\) is significant.

Step 1 — Find the degrees of freedom. \(df = n - 2 = 16 - 2 = 14\).

Step 2 — Set up the critical values. With critical value 0.497, the band runs from \(-0.497\) to \(+0.497\).

Step 3 — Apply the rule. \(r\) is significant only if it lies outside the band. Here \(-0.497 < -0.39 < 0.497\), so \(r\) lies between the critical values.

Step 4 — Conclude. \(r\) is not significant, and the line should not be used for prediction.

Answer: No — since \(-0.39\) lies between \(-0.497\) and \(0.497\), \(r\) is not significant.

Step 1 — Recall how the critical values are positioned. For every degrees-of-freedom row in the table, the critical values come as a matched pair: a negative number and its positive counterpart, e.g. \(-c\) and \(+c\).

Step 2 — Locate zero relative to them. Any pair \(-c\) and \(+c\) straddles zero, so \(r = 0\) always lies strictly between the two critical values.

Step 3 — Apply the rule. The rule says \(r\) is significant only when it lies outside the pair. A value of exactly 0 can never satisfy that condition, no matter how large \(n\) is.

Step 4 — Interpret the result. This makes sense substantively as well: \(r = 0\) means the sample detected no linear association at all, and no amount of extra data turns "no association" into evidence of one.

Answer: \(r = 0\) always falls between the negative and positive critical values, so it can never be significant at any sample size.

Step 1 — Recall what significance establishes. The test shows the linear relationship is unlikely to be a chance artefact of sampling — but it says nothing about the range of \(x\) over which the relationship holds.

Step 2 — Compare the target x-value with the observed domain. The data cover \(x\) from 5 to 25, and \(x = 30\) lies outside that range.

Step 3 — Cite the governing rule. Even when \(r\) is significant and the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed \(x\) values.

Step 4 — Conclude. Predicting \(y\) at \(x = 30\) would be extrapolation and is not justified by the significance test.

Answer: No — although the correlation is significant, \(x = 30\) lies outside the observed domain (5 to 25), so the line may not reliably be used to predict at that point.

Step 1 — Assumption 1: linearity. The relationship between the variables being correlated should be linear; the data points should fall along an approximate straight-line pattern on the scatter plot.

Step 2 — Assumption 2: normality of y about the line. The \(y\) values for any particular \(x\) are normally distributed about the line, with more \(y\) values close to the line than far away; these normal distributions are centred on the line.

Step 3 — Assumption 3: equal standard deviations. The standard deviations of the population \(y\) values about the line are equal for each value of \(x\) — every normal distribution has the same shape and spread about the line.

Step 4 — Assumption 4: independent residuals. The residual errors are mutually independent, showing no pattern.

Step 5 — Assumption 5: random data collection. The data come from a well-designed, random sample or randomized experiment.

Answer: (1) linearity; (2) y-values normally distributed about the line; (3) equal standard deviations of y about the line for all x; (4) mutually independent residual errors; (5) data from a well-designed random sample or randomized experiment.

Step 1 — Name assumption 3 and its check. The residuals plot checks the assumption of equal standard deviations of the \(y\) values about the line (constant error variance). In the plot, this assumption holds when the vertical spread of the residuals stays roughly constant across all values of \(x\); it fails when the residuals fan out (spread widens or narrows) as \(x\) increases.

Step 2 — Name assumption 4 and its check. The residuals plot also checks the assumption that residual errors are mutually independent. This shows up as a plot with no discernible pattern — the points should look randomly scattered above and below zero. A visible pattern (curves, waves, trends) signals dependence and a violated assumption.

Step 3 — Note why this matters. These two assumptions underpin the validity of the significance test, which is why drawing the residuals plot should be routine rather than optional.

Answer: A residuals plot checks assumption 3 (equal spread of y about the line — shown by constant vertical scatter across x) and assumption 4 (independent residuals — shown by no pattern in the plot).