9.2 Confidence Intervals for the Slope of a Regression Model

• In a linear regression model, the slope is often denoted by b_1
• Confidence intervals provide a measure of the uncertainty around the estimated slope.
• If \(b_1 = 2\), it means for every one-unit increase in \(X\), \(Y\) is expected to increase by 2
• The standard error of the slope measures the variability of the estimated slope in a linear regression model.
• What does the standard error of the slope measure in a linear regression model?

  Variability of the estimated slope
• What is the total squared deviations of the X values in the formula for the standard error of the slope?

  $\sum (X_{i} - \bar{X})^{2}$
• The residual standard deviation is calculated as: s
• What is the formula for the degrees of freedom in a linear regression model?

  $df =$$n - 2$
• Match the element with its description:

  Degrees of Freedom (df) ↔️ $n - 2$

  Confidence Level ↔️ Typically 95% or 90%

  Critical Value ↔️ Obtained from t-distribution table
• What does the slope ($b_{1}$) represent in a linear regression model?


  Change in Y for each X
• What do confidence intervals measure in a linear regression model?

  Uncertainty around the estimated slope
• Confidence intervals are a range of plausible values for an unknown parameter
• What does a confidence interval quantify in linear regression?

  The precision of the slope
• The formula for the standard error of the slope is SE(b_1)
• If s = 1.5 and ∑(X_i−Xˉ)² = 20, the standard error of the slope is approximately 0.335
• The standard error of the slope is calculated using the formula: SE(b_1)
• What does \(\sum (X_i - \bar{X})^2\) represent in the formula for the standard error of the slope?

  Total squared deviations of X values
• How are degrees of freedom calculated in a linear regression model?

  $df =$$n - 2$
• What formula is used to calculate the standard error of the slope (\(SE(b_1)\))?

  $\frac{s}{\sqrt{\sum (X_{i} - \bar{X})^{2}}}$
• Degrees of freedom in a linear regression model are calculated as \(n-2\).


  True
• If \(b_1 = 2.5\), \(SE(b_1) = 0.4\), and \(t_{\alpha/2} = 2.086\), the 95% confidence interval for the slope is [1.72, 3.28]
• What does the sign of \(b_1\) indicate in a linear regression model?

  Positive or negative relationship
• What is one purpose of a confidence interval for the slope?

  Quantify the precision
• What is one purpose of a confidence interval for the slope?

  Estimate the true slope
• What is the formula to calculate the standard error of the slope?

  $SE(b_{1}) =$$\frac{s}{\sqrt{\sum (X_{i} - \bar{X})^{2}}}$
• The standard error of the slope is calculated using the formula: s
• If \(s = 1.5\) and $\sum (X_{i} - \bar{X})^{2} =$$20$, then \(SE(b_1) \approx 0.335\)


  True
• What distribution is used to determine critical values for the slope in a linear regression model?

  t-distribution
• Critical values from the t-distribution are denoted as: t_\alpha/2
• What is the primary purpose of confidence intervals for the slope in a linear regression model?

  Estimate the true slope
• The regression equation in a linear regression model is $Y =$$b_{0} +$$b_{1} X +$$\epsilon$


  True
• Purposes of a confidence interval for the slope

  1️⃣ Estimate the true slope in the population

  2️⃣ Quantify the precision of the estimate
• What is the purpose of a confidence interval for the slope in linear regression?

  Estimate the true slope
• A narrow confidence interval indicates a more precise estimate of the slope
• A confidence interval that includes 0 suggests the true slope may be zero, indicating no linear relationship between X and Y.


  True
• In the formula for the standard error of the slope, what does 's' represent?

  Residual standard deviation
• What is the formula for the residual standard deviation 's'?

  s = \sqrt{\frac{\sum (Y_i - \hat{Y}_i)^2}{n - 2}}</latex>
• What is the formula to calculate the standard error of the slope?

  SE(b_1) = \frac{s}{\sqrt{\sum (X_i - \bar{X})^2}}</latex>
• If \(s = 1.5\) and \(\sum (X_i - \bar{X})^2 = 20\), the standard error of the slope is approximately 0.335
• The confidence interval for the slope (\(b_1\)) is calculated as \(b_1 \pm t_{\alpha/2} \cdot SE(b_1)\).

  True