Step 1: Concept Overview:
The problem seeks the critical value of the Pearson correlation coefficient, \(r\), given a sample size and significance level. This involves a t-test to assess the correlation coefficient's significance. The null hypothesis, \(H_0: \rho = 0\), assumes no population correlation, while the alternative hypothesis, \(H_1: \rho eq 0\), suggests otherwise. Significance occurs when the calculated t-statistic exceeds the critical t-value. We aim to determine the smallest \(r\) that satisfies this condition.
Step 2: Core Formula:
The t-statistic for correlation significance is calculated as:
\[ t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}} \]
Where:
- \( r \) represents the sample correlation coefficient.
- \( n \) represents the number of pairs in the sample.
The degrees of freedom (df) are \(n-2\).
Step 3: Step-by-Step Solution:
We have:
- Sample size: \( n = 27 \).
- Significance level: \(\alpha = 0.05\).
- Degrees of freedom: \( df = n - 2 = 27 - 2 = 25 \).
- Critical t-value: \( t_{\text{critical}} = 2.06 \). Correlation significance testing is two-tailed; this value corresponds to \( t_{\alpha/2, df} = t_{0.025, 25} \). The notation \(t_{0.05}(25)\) might be slightly unclear, but 2.06 is the standard critical value for a two-tailed test with \(\alpha = 0.05\) and 25 df.
To find the minimum significant \(r\), equate the calculated t-statistic to the critical t-value and solve for \(r\), considering \(|r|\) for both positive and negative correlations:
\[ \frac{|r| \sqrt{27-2}}{\sqrt{1-r^2}} = 2.06 \]
\[ \frac{|r| \sqrt{25}}{\sqrt{1-r^2}} = 2.06 \]
\[ \frac{5|r|}{\sqrt{1-r^2}} = 2.06 \]
Squaring both sides:
\[ \frac{25r^2}{1-r^2} = (2.06)^2 \]
\[ \frac{25r^2}{1-r^2} = 4.2436 \]
\[ 25r^2 = 4.2436 (1-r^2) \]
\[ 25r^2 = 4.2436 - 4.2436r^2 \]
\[ 25r^2 + 4.2436r^2 = 4.2436 \]
\[ 29.2436r^2 = 4.2436 \]
\[ r^2 = \frac{4.2436}{29.2436} \approx 0.14511 \]
\[ |r| = \sqrt{0.14511} \approx 0.3809 \]
Step 4: Final Result:
The correlation coefficient \(r\) is significant at the 5% level if its absolute value exceeds 0.3809. Hence, the minimum significant positive correlation is approximately 0.381. The condition is \( r>0.381 \).