"[APPLET] For Exercises 1–8, use the data in the table, which shows the average annual salaries (both in thousands of dollars) for secondary and elementary school teachers, excluding special and vocational education teachers, in the United States for 11 years. (Source: U.S. Bureau of Labor Statistics)

8. Construct a 95% prediction interval for the average annual salary of elementary school teachers when the average annual salary of secondary school teachers is \$63,500. Interpret the results."

Verified step by step guidance

1. Step 1: Identify the variables and the goal. Here, x represents the average annual salary of secondary school teachers, and y represents the average annual salary of elementary school teachers. We want to construct a 95% prediction interval for y when x = 63.5 (in thousands of dollars).
2. Step 2: Perform a linear regression analysis to find the regression equation \( \hat{y} = b_0 + b_1 x \). Calculate the slope \( b_1 \) and intercept \( b_0 \) using the formulas: \[ b_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2} \quad \text{and} \quad b_0 = \bar{y} - b_1 \bar{x} \] where \( \bar{x} \) and \( \bar{y} \) are the means of x and y respectively.
3. Step 3: Calculate the predicted value \( \hat{y} \) for \( x = 63.5 \) by substituting into the regression equation: \[ \hat{y} = b_0 + b_1 (63.5) \].
4. Step 4: Compute the standard error of the prediction, which accounts for both the error in estimating the mean response and the variability of individual observations. The formula for the standard error of prediction is: \[ SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(x_0 - \bar{x})^2}{\sum (x_i - \bar{x})^2}} \] where \( s \) is the standard error of the estimate, \( n \) is the number of data points, and \( x_0 = 63.5 \).
5. Step 5: Determine the critical t-value \( t^* \) for a 95% confidence level with \( n-2 \) degrees of freedom. Then construct the 95% prediction interval using: \[ \hat{y} \pm t^* \times SE_{pred} \]. Finally, interpret this interval as the range in which we expect the salary of an individual elementary school teacher to fall when the secondary school teacher salary is \$63,500.

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, with a certain level of confidence (e.g., 95%). It accounts for both the uncertainty in estimating the regression line and the variability of individual data points around that line.

Linear Regression

Linear regression models the relationship between two variables by fitting a straight line that minimizes the sum of squared differences between observed and predicted values. It allows prediction of the dependent variable (elementary salaries) based on the independent variable (secondary salaries).

Interpretation of Confidence and Prediction Intervals

While confidence intervals estimate the mean response for a given predictor value, prediction intervals provide a wider range to predict individual outcomes. Interpreting a 95% prediction interval means we are 95% confident the actual salary will fall within this range for a given secondary teacher salary.