[DATA] Concrete As concrete cures, it gains strength. The following data represent the 7-day and 28-day strength (in pounds per square inch) of a certain type of concrete:

a. Treating the 7-day strength as the explanatory variable, x, determine the estimates of β₀ and β₁.

Verified step by step guidance

1. Step 1: Organize the data by listing all the 7-day strength values as the explanatory variable \(x\) and the corresponding 28-day strength values as the response variable \(y\).
2. Step 2: Calculate the means of \(x\) and \(y\), denoted as \(\bar{x}\) and \(\bar{y}\), using the formulas: \(\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i\) \(\bar{y} = \frac{1}{n} \sum_{i=1}^n y_i\) where \(n\) is the total number of data points.
3. Step 3: Compute the slope estimate \(\hat{\beta}_1\) of the regression line using the formula: \(\hat{\beta}_1 = \frac{\sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^n (x_i - \bar{x})^2}\) This measures how much \(y\) changes for a unit change in \(x\).
4. Step 4: Calculate the intercept estimate \(\hat{\beta}_0\) using the formula: \(\hat{\beta}_0 = \bar{y} - \hat{\beta}_1 \bar{x}\) This represents the expected value of \(y\) when \(x\) is zero.
5. Step 5: Write the estimated regression equation as: \(\hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\) This equation can be used to predict the 28-day strength based on the 7-day strength.

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 and β₁ is the slope, representing the expected change in y for a one-unit change in x. It is used to predict the dependent variable based on the independent variable.

Least Squares Estimation

Least squares estimation is a method to find the best-fitting line by minimizing the sum of the squared differences between observed and predicted values. It provides formulas to calculate estimates of β₀ and β₁ that minimize prediction errors in the regression model.

Explanatory and Response Variables

In regression analysis, the explanatory variable (x) is the predictor or independent variable, while the response variable (y) is the outcome or dependent variable. Understanding which variable explains or predicts the other is crucial for correctly setting up and interpreting the regression.