"Using the sample data from Problem 8 in Section 12.3,
c. Predict the value of y if x=1.8."

Verified step by step guidance

1. Identify the regression equation from Problem 8 in Section 12.3, which should be in the form \(y = \hat{\beta}_0 + \hat{\beta}_1 x\), where \(\hat{\beta}_0\) is the estimated intercept and \(\hat{\beta}_1\) is the estimated slope coefficient.
2. Recall or locate the values of \(\hat{\beta}_0\) and \(\hat{\beta}_1\) from the previous problem or the regression output provided in the textbook.
3. Substitute the given value of \(x = 1.8\) into the regression equation to find the predicted value of \(y\). This means calculating \(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 \times 1.8\).
4. Perform the multiplication and addition as indicated to compute the predicted \(y\) value corresponding to \(x = 1.8\).
5. Interpret the predicted value \(\hat{y}\) as the estimated response variable value when the predictor variable \(x\) is 1.8, based on the linear regression model.

Key Concepts

Simple Linear Regression

Simple linear regression models the relationship between two variables by fitting a straight line to the data. It predicts the dependent variable (y) based on the independent variable (x) using the equation y = b0 + b1*x, where b0 is the intercept and b1 is the slope.

Prediction Using the Regression Equation

Once the regression line is established, predicting y for a given x involves substituting the x-value into the regression equation. This provides an estimated value 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 (slope and intercept). Understanding how these estimates are derived from the sample is essential, as predictions rely on these values to approximate the true relationship in the population.