[DATA] Age versus HDL Cholesterol A doctor wanted to determine whether there is a relation between a male’s age and his HDL (so-called good) cholesterol. He randomly selected 17 of his patients and determined their HDL levels. He obtained the following data.

d. Assuming the residuals are normally distributed, construct a 95% confidence interval for the slope of the true least-squares regression line.

Verified step by step guidance

1. Step 1: Calculate the least-squares regression line for the data. This involves finding the slope (b_1) and intercept (b_0) using the formulas: \(b_1 = \frac{S_{xy}}{S_{xx}}\) where \(S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})\) and \(S_{xx} = \sum (x_i - \bar{x})^2\). Also, \(b_0 = \bar{y} - b_1 \bar{x}\), where \(\bar{x}\) and \(\bar{y}\) are the sample means of \(x\) and \(y\) respectively.
2. Step 2: Calculate the residual standard error (also called the standard error of the estimate), \(s\), which measures the typical distance that the observed values fall from the regression line. Use the formula: \(s = \sqrt{\frac{1}{n-2} \sum (y_i - \hat{y}_i)^2}\), where \(\hat{y}_i = b_0 + b_1 x_i\) are the predicted values from the regression line, and \(n\) is the number of data points.
3. Step 3: Calculate the standard error of the slope, \(SE_{b_1}\), which quantifies the variability of the slope estimate. Use the formula: \(SE_{b_1} = \frac{s}{\sqrt{S_{xx}}}\).
4. Step 4: Determine the critical t-value, \(t^*\), for a 95% confidence interval with \(n-2\) degrees of freedom. This value comes from the t-distribution table and depends on the sample size.
5. Step 5: Construct the 95% confidence interval for the slope using the formula: \(b_1 \pm t^* \times SE_{b_1}\). This interval estimates the range of plausible values for the true slope of the regression line, assuming the residuals are normally distributed.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data points by minimizing the sum of the squared differences between observed and predicted values. It models the relationship between an independent variable (age) and a dependent variable (HDL cholesterol), allowing predictions and interpretation of the slope as the average change in HDL per unit change in age.

Residuals and Their Normality

Residuals are the differences between observed values and the values predicted by the regression line. Assuming residuals are normally distributed is crucial for valid inference, such as constructing confidence intervals, because it ensures the sampling distribution of the slope estimator follows a t-distribution, enabling accurate hypothesis testing and interval estimation.

Confidence Interval for the Slope

A confidence interval for the slope estimates the range within which the true slope of the population regression line lies with a specified level of confidence (e.g., 95%). It is calculated using the slope estimate, its standard error, and the t-distribution critical value, providing insight into the strength and direction of the relationship between variables.