To determine whether a correlation exists between two variables at a significance level of \( \alpha = 0.01 \), you'll typically follow these steps:
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State the Hypotheses:
- Null Hypothesis (\( H_0 \)): There is no correlation between the two variables (e.g., \( \rho = 0 \)).
- Alternative Hypothesis (\( H_a \)): There is a correlation between the two variables (e.g., \( \rho \neq 0 \)).
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Collect Data: Collect paired data observations for the two variables you are examining.
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Calculate the Correlation Coefficient: Use Pearson's correlation coefficient formula to compute \( r \).
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Determine the Critical Value: Use a correlation significance table or a calculator to find the critical values for \( \alpha = 0.01 \) based on your sample size. The critical value can be found for \( n - 2 \) degrees of freedom, where \( n \) is the number of pairs.
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Calculate the Test Statistic:
- Convert the correlation coefficient \( r \) into a test statistic using the formula: \[ t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}} \] where \( n \) is the sample size.
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Compare the Test Statistic to Critical Values:
- If the absolute value of the calculated \( t \) is greater than the critical value from the t-distribution, you reject the null hypothesis.
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Draw a Conclusion:
- If you reject \( H_0 \), it suggests that there is a significant correlation between the variables. Otherwise, you do not have enough evidence to suggest a correlation exists.
If you have specific data, I can help you through each step with actual calculations. Please provide the data or the correlation coefficient and sample size, if available.