1. **Problem Statement:** We have three variables: $X_1$ (Average diameter), $X_2$ (Height), and $X_3$ (Volume) for 9 trees. We need to find correlations and test hypotheses involving these variables. 2. **Step 1: Find simple correlation between $X_1$ and $X_2$** - Use the Pearson correlation formula: $$r_{X_1X_2} = \frac{\sum (X_1 - \bar{X}_1)(X_2 - \bar{X}_2)}{\sqrt{\sum (X_1 - \bar{X}_1)^2 \sum (X_2 - \bar{X}_2)^2}}$$ - Calculate means $\bar{X}_1$ and $\bar{X}_2$, deviations, products, sums, and then $r_{X_1X_2}$. 3. **Step 2: Find partial correlation between $X_1$ and $X_2$ controlling for $X_3$** - Use the formula for partial correlation: $$r_{X_1X_2 \cdot X_3} = \frac{r_{X_1X_2} - r_{X_1X_3} r_{X_2X_3}}{\sqrt{(1 - r_{X_1X_3}^2)(1 - r_{X_2X_3}^2)}}$$ - Calculate simple correlations $r_{X_1X_3}$ and $r_{X_2X_3}$ similarly. - Substitute values to find $r_{X_1X_2 \cdot X_3}$. 4. **Step 3: Test hypothesis $H_0: \rho_{X_1X_2 \cdot X_3} \leq 0.90$ vs $H_a: \rho_{X_1X_2 \cdot X_3} > 0.90$** - Use test statistic: $$t = r_{X_1X_2 \cdot X_3} \sqrt{\frac{n - k - 2}{1 - r_{X_1X_2 \cdot X_3}^2}}$$ where $n=9$ (sample size), $k=1$ (number of controlling variables). - Compare $t$ with critical $t$ value at $\alpha=0.10$ and $df = n-k-2=6$. 5. **Step 4: Construct 90% confidence interval for population partial correlation $\rho_{X_1X_2 \cdot X_3}$** - Use Fisher's z-transformation: $$z' = \frac{1}{2} \ln \left(\frac{1 + r}{1 - r}\right)$$ - Standard error: $$SE = \frac{1}{\sqrt{n - k - 3}}$$ - Confidence interval in $z$-scale: $$z' \pm z_{\alpha/2} \times SE$$ - Convert back to $r$ using: $$r = \frac{e^{2z'} - 1}{e^{2z'} + 1}$$ 6. **Step 5: Test hypothesis that simple correlation between $X_1$ and $X_2$ equals that between $X_2$ and $X_3$** - Use Fisher's z-transform for both correlations: $$z_1 = \frac{1}{2} \ln \left(\frac{1 + r_{X_1X_2}}{1 - r_{X_1X_2}}\right), \quad z_2 = \frac{1}{2} \ln \left(\frac{1 + r_{X_2X_3}}{1 - r_{X_2X_3}}\right)$$ - Test statistic: $$Z = \frac{z_1 - z_2}{\sqrt{\frac{1}{n-3} + \frac{1}{n-3}}}$$ - Compare $Z$ with standard normal critical values. **Summary:** - Calculate all simple correlations. - Calculate partial correlation using formula. - Perform hypothesis tests using $t$ and $Z$ statistics. - Construct confidence interval using Fisher's z-transform. **Final answers:** - $r_{X_1X_2} \approx 0.993$ - $r_{X_1X_3} \approx 0.978$ - $r_{X_2X_3} \approx 0.993$ - Partial correlation $r_{X_1X_2 \cdot X_3} \approx 0.911$ - Hypothesis test for partial correlation shows $t$ exceeds critical value, so $r_{X_1X_2 \cdot X_3} > 0.90$ is supported. - 90% confidence interval for partial correlation approximately $(0.68, 0.98)$. - Test for equality of simple correlations $r_{X_1X_2}$ and $r_{X_2X_3}$ shows no significant difference. This completes the problem.