[NOW WORK] Using the sample data from Problem 5 in Section 12.3,
a. Predict the mean value of y if x = 7.

Verified step by step guidance

1. Identify the regression equation from Problem 5 in Section 12.3, which is typically in the form \(y = b_0 + b_1 x\), where \(b_0\) is the intercept and \(b_1\) is the slope of the regression line.
2. Substitute the given value of \(x = 7\) into the regression equation to find the predicted mean value of \(y\).
3. Write the expression for the predicted mean value as \(\hat{y} = b_0 + b_1 \times 7\).
4. Use the values of \(b_0\) and \(b_1\) from the sample data provided in Problem 5 to calculate \(\hat{y}\).
5. Interpret the result as the estimated average value of \(y\) when \(x\) is 7, based on the linear relationship modeled by the regression equation.

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 given 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 from a previous problem is used to estimate the regression parameters (intercept and slope). Understanding how these estimates are derived is essential to apply the regression equation correctly for prediction.