Using the sample data from Problem 6 in Section 12.3,
a. Predict the mean value of y if x=8.

Verified step by step guidance

1. Identify the regression equation from Problem 6 in Section 12.3, which is typically in the form \(\hat{y} = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope of the regression line.
2. Locate the values of the intercept \(b_0\) and slope \(b_1\) from the given sample data or previous calculations in Problem 6.
3. Substitute the given value of \(x = 8\) into the regression equation to find the predicted mean value of \(y\). This means calculating \(\hat{y} = b_0 + b_1 \times 8\).
4. Perform the multiplication and addition as indicated to express the predicted mean value of \(y\) in terms of \(b_0\) and \(b_1\).
5. Interpret the result as the estimated mean response of \(y\) when \(x\) is 8, based on the linear relationship established by the regression model.

Key Concepts

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 equation y = b0 + b1*x estimates the mean value of y for any given x, where b0 is the intercept and b1 is the slope.

Prediction Using the Regression Equation

To predict the mean value of y for a specific x, substitute the x-value into the regression equation. This provides an estimated average response of y based on the linear relationship derived from the sample data.

Sample Data and Parameter Estimation

Sample data are used to estimate the regression parameters (intercept and slope). These estimates allow us to make predictions about the population mean response, assuming the linear model fits the data well.