Question

a. Explain the difference between the line \(y=\) \(\alpha+\beta x\) and the line \(\hat{y}=a+b x\). b. Explain the difference between \(\beta\) and \(b\). c. Let \(x^{*}\) denote a particular value of the independent variable. Explain the difference between \(\alpha+\beta x^{*}\) and \(a+b x^{*}\) d. Explain the difference between \(\sigma\) and \(s_{e}\).

Short answer

a. \(y = \alpha + \beta x\) is population regression line while \(\hat{y}=a+bx\) is sample regression line.\nb. \(\beta\) is population slope coefficient while \(b\) is sample slope coefficient.\nc. \(\alpha + \beta x^{*}\) is population mean value of \(y\) given \(x^{*}\); \(a+b x^{*}\) is predicted value of \(y\) for \(x^{*}\) based on sample data.\nd. \(\sigma\) is standard deviation in population; \(s_{e}\) is standard error of the estimate.

Step-by-step solution

Difference between \(y = \alpha + \beta x\) and \(\hat{y} = a + bx\)

Line \(y = \alpha + \beta x\) represents population regression line which shows the true relationship between the dependent variable \(y\) and the independent variable \(x\) in the entire population.\nLine \(\hat{y} = a + bx\) represents sample regression line which shows the estimated relationship between the dependent variable \(y\) and the independent variable \(x\) in a sample.

Explain the difference between \(\beta\) and \(b\)

\(\beta\) is the population slope coefficient. It represents the change in the mean of the dependent variable for one unit of change in the independent variable.\n\(b\) is the sample slope coefficient. It is the best estimate of \(\beta\) based on sample data. It is calculated using the least squares estimation method from the sample data.

Difference between \(\alpha + \beta x^{*}\) and \(a + b x^{*}\)

\(\alpha + \beta x^{*}\) represents the population mean value of \(y\) given a particular value of \(x\), \(x^{*}\).\n \(a + b x^{*}\) is the predicted or expected value of \(y\) for a particular value of \(x\), \(x^{*}\), based on the sample data.

Difference between \(\sigma\) and \(s_{e}\)

\(\sigma\) represents standard deviation in the population. It measures the variation or dispersion of the entire population data from the mean.\nMeanwhile, \(s_{e}\) is standard error of the estimate. It indicates the spread of the distribution of errors. It is a measure of the accuracy of predictions made with a regression line.