Understanding P-value Significance for Correlation Coefficients

In statistics, the p-value is a crucial concept used in hypothesis testing. When we calculate a sample correlation coefficient, we often want to know if this observed correlation is statistically significant, meaning it is unlikely to have occurred just by random chance if there were truly no correlation in the population.

The significance level, denoted by the Greek letter α (alpha), is a threshold probability that we set before conducting a test. Common significance levels are 5% (0.05), 1% (0.01), and 10% (0.10).

The decision rule for hypothesis testing using the p-value is:

• If the p-value is less than or equal to α ($\text{p-value} \le \alpha$), we reject the null hypothesis. This means the result (e.g., the observed correlation) is considered statistically significant at the chosen α level.
• If the p-value is greater than α ($\text{p-value} > \alpha$), we do not reject the null hypothesis. This means the result is not considered statistically significant at the chosen α level.

Applying the Rule to 1% Significance Level

The question asks for a p-value that signifies significance at the 1% level of significance. This means our significance level α is 1%, which is equivalent to 0.01 in decimal form ($\alpha = 0.01$).

We are looking for a p-value from the options provided that is less than or equal to 0.01 ($\text{p-value} \le 0.01$).

Evaluating the Options

Let's examine each given p-value option:

• Option 1: p-value = 0.005
• Option 2: p-value = 0.02
• Option 3: p-value = 0.99
• Option 4: p-value = 0.95

Now, we compare each p-value with our significance level $\alpha = 0.01$:

From the evaluation, only the p-value of 0.005 is less than or equal to 0.01. Therefore, a sample correlation coefficient yielding a p-value of 0.005 would be considered significant at the 1% level of significance.

Revision Table: Key Concepts

Additional Information: P-values and Correlation Testing

When testing for the significance of a sample correlation coefficient (like Pearson's r), the null hypothesis ($H_0$) typically states that the true population correlation coefficient ($\rho$) is zero. The alternative hypothesis ($H_1$) usually states that the true population correlation is not zero ($\rho \neq 0$), or is positive ($\rho > 0$), or is negative ($\rho < 0$).

Statistical software calculates the sample correlation coefficient and then computes a test statistic (often a t-statistic) based on the sample size and the coefficient value. This test statistic is then used to find the p-value, which represents the probability of observing a correlation as strong as, or stronger than, the one found in the sample, assuming the true correlation in the population is zero.

A small p-value (like 0.005) suggests that it would be very unusual to observe such a strong correlation if the true correlation were zero. This provides strong evidence against the null hypothesis ($H_0: \rho = 0$), leading us to reject $H_0$ and conclude that the observed correlation is statistically significant at the chosen α level. A p-value like 0.99, on the other hand, means that observing the sample correlation is very likely even if the true population correlation is zero, providing no evidence to reject $H_0$.