[DATA] Calories versus Sugar The following data represent the number of calories per serving and the number of grams of sugar per serving for a random sample of high-protein and moderate-protein energy bars.

g. If the residuals are normally distributed, construct a 95% confidence interval about 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) and intercept (a) using the formulas: \[ b = \frac{S_{xy}}{S_{xx}} \] \[ a = \bar{y} - b\bar{x} \] where \(S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y})\) and \(S_{xx} = \sum (x_i - \bar{x})^2\), and \(\bar{x}\) and \(\bar{y}\) are the sample means of \(x\) and \(y\) respectively.
2. Step 2: Calculate the residuals for each data point, which are the differences between the observed values \(y_i\) and the predicted values \(\hat{y}_i\) from the regression line: \[ e_i = y_i - \hat{y}_i \]
3. Step 3: Compute the standard error of the slope, \(SE_b\), using the formula: \[ SE_b = \frac{s}{\sqrt{S_{xx}}} \] where \(s\) is the standard error of the estimate, calculated as \[ s = \sqrt{\frac{\sum e_i^2}{n-2}} \] with \(n\) being the number of data points.
4. Step 4: Determine the critical value \(t^*\) from the \(t\)-distribution with \(n-2\) degrees of freedom for a 95% confidence level. This value corresponds to the two-tailed probability of 0.05.
5. Step 5: Construct the 95% confidence interval for the slope \(b\) using the formula: \[ b \pm t^* \times SE_b \] This interval estimates the range of plausible values for the true slope of the regression line.

Key Concepts

Least-Squares Regression Line

The least-squares regression line is a straight line that best fits the data by minimizing the sum of the squared differences between observed and predicted values. It models the relationship between an independent variable (x) and a dependent variable (y), allowing predictions of y based on x.

Residuals and Their Distribution

Residuals are the differences between observed values and the values predicted by the regression line. Assuming residuals are normally distributed is crucial for making valid inferences, such as constructing confidence intervals, because it ensures the reliability of statistical tests and estimates.

Confidence Interval for the Slope

A confidence interval for the slope estimates the range within which the true slope of the regression line lies with a specified level of confidence (e.g., 95%). It quantifies the uncertainty in the slope estimate and is calculated using the standard error of the slope and the t-distribution.