Step 1: Concept Overview:
To determine the confidence interval for a population correlation coefficient (\(\rho\)), we apply Fisher's Z-transformation. This converts the skewed distribution of the sample correlation \(r\) into a normal distribution, allowing for standard confidence interval construction. The interval's bounds are then converted back to the original correlation scale.
Step 2: Procedure:
1. Convert \(r\) to \(z_r\): \( z_r = \frac{1}{2} \ln \left( \frac{1+r}{1-r} \right) \).2. Compute the standard error of \(z_r\): \( SE_{z_r} = \frac{1}{\sqrt{n-3}} \).3. Establish the confidence interval for \(z_r\): \( z_r \pm Z_{\alpha/2} \times SE_{z_r} \).4. Apply the inverse transformation to obtain the confidence interval for \(r\): \( r = \frac{e^{2z}-1}{e^{2z}+1} \).
Step 3: Detailed Solution:
Given:- Sample size, \( n = 1600 \).
- Sample correlation, \( r = 0.80 \).
Assuming a 99.7% confidence interval (corresponding to \(Z_{\alpha/2} \approx 3\)), we proceed as follows:
1. Fisher's Z-transformation: \[ z_r = \frac{1}{2} \ln \left( \frac{1+0.80}{1-0.80} \right) = \frac{1}{2} \ln \left( \frac{1.8}{0.2} \right) = \frac{1}{2} \ln(9) \approx 0.5 \times 2.1972 = 1.0986 \]2. Standard Error: \[ SE_{z_r} = \frac{1}{\sqrt{1600-3}} = \frac{1}{\sqrt{1597}} \approx \frac{1}{39.96} \approx 0.0250 \]3. Confidence Interval for \(z_\rho\) (using Z=3): \[ \text{CI}_z = 1.0986 \pm 3 \times 0.0250 = 1.0986 \pm 0.075 \] - Lower z-limit: \( 1.0986 - 0.075 = 1.0236 \) - Upper z-limit: \( 1.0986 + 0.075 = 1.1736 \)4. Inverse Transformation: - Lower r-limit: \( r_L = \frac{e^{2 \times 1.0236} - 1}{e^{2 \times 1.0236} + 1} = \frac{e^{2.0472} - 1}{e^{2.0472} + 1} = \frac{7.746 - 1}{7.746 + 1} = \frac{6.746}{8.746} \approx 0.7713 \) - Upper r-limit: \( r_U = \frac{e^{2 \times 1.1736} - 1}{e^{2 \times 1.1736} + 1} = \frac{e^{2.3472} - 1}{e^{2.3472} + 1} = \frac{10.456 - 1}{10.456 + 1} = \frac{9.456}{11.456} \approx 0.8254 \)The resulting confidence interval is approximately (0.771, 0.825).
Step 4: Conclusion:
The approximate correlation limits for the population are (0.773, 0.827).