Answer: We can be 95% confident that the value of Y will increase by between 20 and 25 units for every one unit increase in X

## Explanation In regression analysis, β₁ represents the slope coefficient, which indicates the change in the dependent variable Y for a one-unit change in the independent variable X. ### Key Points: 1. **β₁ Interpretation**: β₁ is the regression coefficient that measures the relationship between X and Y. Specifically, it tells us how much Y changes when X increases by one unit. 2. **Confidence Interval Interpretation**: A 95% confidence interval for β₁ of (20, 25) means we are 95% confident that the true population slope parameter β₁ lies between 20 and 25. 3. **Practical Meaning**: Since β₁ = ΔY/ΔX, this confidence interval tells us that for every one-unit increase in X, we can be 95% confident that Y will increase by between 20 and 25 units. ### Why Other Options Are Incorrect: - **Option A**: This describes a confidence interval for the mean value of Y, not for the slope coefficient β₁. - **Option B**: This reverses the relationship - β₁ measures the change in Y for a change in X, not the change in X for a change in Y. - **Option D**: A confidence interval that does not contain zero (20 to 25 is entirely positive) actually suggests evidence of a linear relationship at the 5% significance level, not the absence of one. ### Statistical Context: The confidence interval (20, 25) being entirely positive indicates a statistically significant positive relationship between X and Y at the 5% significance level (since 0 is not within the interval).

Author: Nikitesh Somanthe