Final answer:
The correlation coefficient r = -0.036 for a sample size of n = 25 is not statistically significant at the α = 0.05 level because it does not exceed the corresponding critical value of approximately ±0.396, indicating that there is not a significant linear relationship between the two variables.
Step-by-step explanation:
Determining the Significance of a Correlation Coefficient
To determine whether a given correlation coefficient r = -0.036 is statistically significant for a sample size of n = 25 at a significance level of α = 0.05, we first identify the critical value that corresponds to our degrees of freedom (df). The degrees of freedom can be calculated by taking the sample size and subtracting 2 (df = n - 2). For our case, df = 25 - 2 = 23. We would then consult a table of critical values or use statistical software to find the critical value corresponding to df = 23 at the 0.05 significance level.
According to such a table, the critical value for a two-tailed test at α = 0.05 and df = 23 is approximately ±0.396. Since our correlation coefficient of -0.036 does not exceed the absolute value of the critical value (i.e., it is not less than -0.396 or greater than 0.396), we fail to reject the null hypothesis. This indicates that the correlation is not statistically significant, and we would conclude that there is not sufficient evidence to suggest a significant linear relationship between the two variables at the α = 0.05 level.
When interpreting correlation coefficients, it is crucial to consider both the value of r and its significance. A correlation coefficient close to 0 indicates no linear relationship, while a coefficient closer to -1 or 1 indicates a stronger negative or positive linear relationship, respectively. However, without statistical significance, it does not provide enough evidence to suggest that the relationship is real rather than a result of random chance in the sample data.