he Pearson correlation coefficient for a relationship between two variables from a data sample of n=9
was found to be r≈−0.676
. Is this relationship statistically significant at the α=0.05
level?
Critical Values of the Pearson Correlation Coefficient
n	α=0.05
α=0.01
4
0.950
0.990
5
0.878
0.959
6
0.811
0.917
7
0.754
0.875
8
0.707
0.834
9
0.666
0.798
10
0.632
0.765
11
0.602
0.735
12
0.576
0.708

To determine if the relationship between the two variables is statistically significant at the \(\alpha = 0.05\) level, we need to compare the absolute value of the Pearson correlation coefficient \( |r| \) with the critical value for \( n = 9 \).

Given:
- Sample size \( n = 9 \)
- Pearson correlation coefficient \( r \approx -0.676 \)
- Critical value for \( n = 9 \) at \( \alpha = 0.05 \) is \( 0.666 \)

The absolute value of \( r \) is \( |r| = 0.676 \).

Since \( 0.676 \) is greater than the critical value \( 0.666 \) at the \( \alpha = 0.05 \) level, the correlation coefficient is statistically significant.

Therefore, the relationship between the two variables is statistically significant at the \(\alpha = 0.05\) level.

Would you like more details or have any questions?

Here are 5 questions you might ask:
1. How is the Pearson correlation coefficient calculated?
2. What does it mean for a correlation to be statistically significant?
3. Can the Pearson correlation coefficient be used for non-linear relationships?
4. How do different sample sizes affect the critical values of the Pearson correlation coefficient?
5. What are some limitations of using the Pearson correlation coefficient?

**Tip:** The closer the absolute value of the Pearson correlation coefficient is to 1, the stronger the linear relationship between the variables.

Explore the statistical significance of the Pearson correlation coefficient r ≈ -0.676 with a detailed analysis. Understand how to determine significance levels and critical values for different sample sizes. Suitable for advanced college students and researchers.

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