Hypothesis Tests for Zero Correlation

Pearson's $r$ lies between $-1$ and $1$ and measures how closely paired data follow a straight line. The sign gives direction. A value near zero means little linear association; it does not rule out a curved association.

$r$ is a sample statistic. $\rho$ is the population linear correlation coefficient. The usual null is $H_0:\rho=0$, not $r=0$. The observed $r$ is already known and is not the unknown claim being tested.

OCR H240 provides or lets you use the correlation coefficient and critical values; calculation of the coefficient itself is excluded. Concentrate on interpretation, selecting the correct reference and giving a justified decision.

Use $H_1:\rho>0$ for a predicted positive association, $H_1:\rho<0$ for a predicted negative one, and $H_1:\rho\neq0$ if either direction is relevant.

Use the given sample size and significance level when reading the table. Inspect whether the heading gives one-tailed levels, two-tailed levels, or both. A 5% two-tailed test places 2.5% in each tail, but a table may already call that column “5% two-tailed”.

For a positive alternative, reject only when the observed r is sufficiently positive. For a negative alternative, reject only when it is sufficiently negative. Taking the absolute value without checking direction can reverse a conclusion.

For a two-tailed test with supplied critical magnitude $c$, reject for $r<-c$ or $r>c$, equivalently $|r|>c$. Both signs count because both conflict with $\rho=0$ in the stated alternative.

For a one-tailed positive alternative with critical magnitude $c$, a large negative $r$ is not in the rejection region. Do not change to a negative or two-sided alternative after seeing the data.

If a p-value is provided for the stated alternative, compare it directly with the significance level. Do not halve or double it unless the question explicitly gives a probability for a different tail arrangement and the appropriate conversion is justified.

The standard Pearson correlation test assumes the paired observations arise from a bivariate normal population; random independent pairs provide the sampling basis. Inspect a scatter plot for outliers, subgroups and curvature when information is available.

An unusual point can dominate the coefficient. Two distinct subgroups can produce an association that differs from the relationship inside either subgroup. Interpret the actual sampling frame rather than extending the result to everyone.

Evidence of correlation does not establish a causal link. A third variable, reverse direction or selection effects can explain an association. A larger sample can detect a weak correlation that has little practical predictive value.