[NOW WORK] Using the sample data from Problem 5 in Section 12.3,
d. Construct a 95% prediction interval for the value of y if x=7.

Verified step by step guidance

1. Identify the regression equation from the sample data in Problem 5 of Section 12.3. This equation is typically of the form \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope.
2. Calculate the predicted value \(\hat{y}\) at \(x=7\) by substituting \(x=7\) into the regression equation: \(\hat{y} = b_0 + b_1 \times 7\).
3. Find the standard error of the prediction, which accounts for both the variability of the estimate of the mean response and the variability of individual observations. The formula for the standard error of prediction is: \[ SE_{pred} = s \sqrt{1 + \frac{1}{n} + \frac{(7 - \bar{x})^2}{\sum (x_i - \bar{x})^2}} \] where \(s\) is the standard error of the estimate, \(n\) is the sample size, \(\bar{x}\) is the mean of the \(x\) values, and \(x_i\) are the individual \(x\) values.
4. Determine the critical value from the \(t\)-distribution for a 95% prediction interval with \(n-2\) degrees of freedom. This value is denoted as \(t_{\alpha/2, n-2}\), where \(\alpha = 0.05\).
5. Construct the 95% prediction interval for \(y\) at \(x=7\) using the formula: \[ \hat{y} \pm t_{\alpha/2, n-2} \times SE_{pred} \] This interval gives the range in which a new individual observation of \(y\) is expected to fall with 95% confidence when \(x=7\).

Key Concepts

Prediction Interval

A prediction interval estimates the range within which a single new observation is expected to fall, given a specific value of the predictor variable. Unlike confidence intervals for the mean response, prediction intervals account for both the uncertainty in estimating the mean and the variability of individual observations.

Simple Linear Regression

Simple linear regression models the relationship between a dependent variable y and an independent variable x by fitting a straight line. The model estimates parameters to predict y based on x, which is essential for constructing prediction intervals at specific x values.

Standard Error of Prediction

The standard error of prediction measures the variability of predicted values around the true value for a new observation. It combines the error in estimating the regression line and the inherent variability of data points, and is used to calculate the width of the prediction interval.