In Problems 5–10, use the results of Problems 7–12, respectively, from Section 4.2 to answer the following questions:


a. What are the estimates of β₀ and β₁?

Verified step by step guidance

1. Step 1: Identify the data points given. Here, the paired values are (x, y) = (3, 4), (4, 6), (5, 7), (7, 12), and (8, 14).
2. Step 2: Calculate the means of x and y. Use the formulas: \(\bar{x} = \frac{\sum x_i}{n}\) and \(\bar{y} = \frac{\sum y_i}{n}\), where \(n\) is the number of data points.
3. Step 3: Calculate the slope \(\beta_1\) of the regression line using the formula: \(\beta_1 = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}\).
4. Step 4: Calculate the intercept \(\beta_0\) using the formula: \(\beta_0 = \bar{y} - \beta_1 \bar{x}\).
5. Step 5: Interpret \(\beta_0\) as the estimated value of \(y\) when \(x=0\), and \(\beta_1\) as the estimated change in \(y\) for a one-unit increase in \(x\).

Key Concepts

Simple Linear Regression

Simple linear regression models the relationship between two variables by fitting a linear equation y = β₀ + β₁x. Here, β₀ is the intercept, representing the predicted value of y when x = 0, and β₁ is the slope, indicating the change in y for a one-unit increase in x.

Estimation of Regression Coefficients (β₀ and β₁)

The coefficients β₀ and β₁ are estimated using the least squares method, which minimizes the sum of squared differences between observed and predicted y-values. β₁ is calculated as the covariance of x and y divided by the variance of x, while β₀ is found by subtracting β₁ times the mean of x from the mean of y.

Interpretation of Paired Data in Regression

Paired data (x, y) are used to determine the linear relationship between variables. Understanding how to organize and analyze these pairs is essential for calculating means, variances, and covariances, which are necessary steps in estimating regression coefficients and interpreting the model.