The sample regression line a. is used to estimate the population regression line. b. is estimated by the population regression line. c. connects the data points in a sample. d. maximizes the sum of the squared differences between the data points in a sample and the sample regression line. e. shows the actual (or true) relation between the dependent and independent variables.
The correct answer and explanation is:
The correct answer is a. is used to estimate the population regression line.
Explanation:
A sample regression line is a statistical tool used in linear regression to estimate the relationship between a dependent variable and an independent variable in a sample of data. The population regression line represents the true relationship between the variables in the entire population, but since data from the whole population is often unavailable, the sample regression line is used as an approximation.
The equation of the sample regression line is determined by the method of least squares, which minimizes the sum of the squared differences between the observed data points and the values predicted by the regression line. This approach gives the best-fitting line through the sample data points, providing estimates for the slope and intercept of the population regression line. These estimates are used to make inferences about the population relationship between the variables.
Option b is incorrect because the sample regression line is not estimated by the population regression line; rather, the population regression line is what we aim to estimate using the sample regression line.
Option c is incorrect because the sample regression line does not connect the data points directly; it provides a model for the general trend in the data.
Option d is incorrect because the method of least squares minimizes the sum of the squared residuals (differences between the observed values and the predicted values), not maximizes it.
Option e is also incorrect because the sample regression line is an estimate of the true relationship in the population, not the actual relationship itself.
Thus, the sample regression line is primarily used to estimate the population regression line, making option a the correct answer.