Final answer:
The statistics a and b calculated from the least squares regression line are designed to be unbiased estimators of the population parameters alpha and beta. They provide the best-fit line for sample data and, through minimizing the sum of squared errors, can estimate the population standard deviation of y when accompanied by significant correlation.
Step-by-step explanation:
When we calculate the least squares regression line, the statistics a (intercept) and b (slope) are indeed meant to be unbiased estimators of the true population parameters α (alpha) and β (beta), respectively. These estimators are derived from sample data and are used to make inferences about the population parameters. By definition, an unbiased estimator is expected to equal the parameter it estimates over many samples.
The least squares method minimizes the sum of squared errors (SSE) to find the best-fit line described by the equation ŷ = a + bx. The line represents the expected value of y, given x, in our sample data. To confirm that the regression line is a good fit for the data, we look for a significant correlation coefficient, examine scatter plots for a visual relationship, and may seek to minimize residuals to ensure the line of best fit accurately represents the data.