Q: “Test the significance of the correlation coefficient using a t-test at a significance level of 5%”
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To test the significance of a correlation coefficient, a t-test can be used to determine if the observed correlation is statistically significant. This involves testing the null hypothesis that there is no correlation between the two variables in the population (i.e., the correlation coefficient is zero) against the alternative hypothesis that there is a significant correlation.
Steps to Perform the t-Test:
- Calculate the Correlation Coefficient ®:
First, compute the Pearson correlation coefficient for your sample data. This coefficient measures the strength and direction of the linear relationship between two variables. - Determine the Sample Size (n):
Identify the number of paired observations in your sample. - Calculate the t-Statistic:
Use the following formula to calculate the t-statistic:
[
T = \frac{r \sqrt{n – 2}}{\sqrt{1 – r^2}}
]
Where ( r ) is the correlation coefficient and ( n ) is the sample size. - Determine the Degrees of Freedom (df):
The degrees of freedom for the test is ( n – 2 ). - Find the Critical Value:
Using a t-distribution table or statistical software, find the critical t-value for the given significance level (5%) and degrees of freedom. - Compare the t-Statistic to the Critical Value:
Compare the calculated t-statistic with the critical t-value. If the absolute value of the t-statistic exceeds the critical value, reject the null hypothesis.
Interpretation:
- Rejecting the Null Hypothesis: If the t-statistic is greater than the critical value, it indicates that the correlation coefficient is significantly different from zero at the 5% significance level. This means there is evidence of a significant linear relationship between the variables.
- Failing to Reject the Null Hypothesis: If the t-statistic is less than the critical value, there is not enough evidence to conclude that the correlation coefficient is significantly different from zero. Thus, the observed correlation could be due to random chance.
This process allows researchers to assess whether the observed correlation in a sample is statistically significant, helping to determine if it likely reflects a true relationship in the population or if it might have occurred by chance.