Variation and Prediction Intervals
In Exercises 17–20, find the (a) explained variation, (b) unexplained variation, and (c) indicated prediction interval. In each case, there is sufficient evidence to support a claim of a linear correlation, so it is reasonable to use the regression equation when making predictions.
Altitude and Temperature Listed below are altitudes (thousands of feet) and outside air temperatures (°F) recorded by the author during Delta Flight 1053 from New Orleans to Atlanta. For the prediction interval, use a 95% confidence level with the altitude of 6327 ft (or 6.327 thousand feet).

Verified step by step guidance

1. Step 1: Begin by calculating the regression equation (y = mx + b) using the given data. To do this, compute the slope (m) and y-intercept (b) using the formulas: m = (Σ(xy) - n(μx)(μy)) / (Σ(x^2) - n(μx^2)) and b = μy - mμx, where μx and μy are the means of x (altitude) and y (temperature).
2. Step 2: Compute the explained variation (SS_reg). This is the sum of squared differences between the predicted values (ŷ) and the mean of the observed values (μy). Use the formula: SS_reg = Σ(ŷ - μy)^2.
3. Step 3: Compute the unexplained variation (SS_res). This is the sum of squared differences between the observed values (y) and the predicted values (ŷ). Use the formula: SS_res = Σ(y - ŷ)^2.
4. Step 4: Calculate the total variation (SS_tot) as the sum of explained and unexplained variations: SS_tot = SS_reg + SS_res. Verify that this relationship holds true.
5. Step 5: For the prediction interval at an altitude of 6327 ft (6.327 thousand feet), use the regression equation to predict the temperature (ŷ). Then, calculate the margin of error using the standard error of the estimate and the t-value for a 95% confidence level. The prediction interval is given by: ŷ ± margin of error.

Key Concepts

Explained Variation

Explained variation refers to the portion of the total variation in the dependent variable (in this case, temperature) that can be attributed to the independent variable (altitude). It is calculated using the regression model, where the sum of squares of the regression (SSR) indicates how much of the variation is explained by the model. A higher explained variation suggests a stronger relationship between the variables.

Unexplained Variation

Unexplained variation, also known as residual variation, is the part of the total variation in the dependent variable that cannot be accounted for by the independent variable. It is represented by the sum of squares of the residuals (SSE) in a regression analysis. Understanding unexplained variation is crucial for assessing the accuracy of predictions made by the regression model, as it indicates the degree of error in the predictions.

Prediction Interval

A prediction interval provides a range of values within which we expect a future observation to fall, given a certain level of confidence (e.g., 95%). It takes into account both the variability of the data and the uncertainty in the regression model. The prediction interval is wider than the confidence interval for the mean response because it includes the additional variability of individual observations, making it essential for making informed predictions.