The regression model is [math]y =\beta_0 + \beta_1x + \epsilon.[/math] There are six observations.

The summary statistics are:

[[math]] \sum y_i = 8.5, \, \sum x_i = 6, \, \sum x_i^2 = 16, \sum x_iy_i = 15.5, \, \sum y_i^2 = 17.25. [[/math]]

Calculate the least squares estimate of [math]\beta_1[/math].

• 0.1
• 0.3
• 0.5
• 0.7
• 0.9

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Key: D

[[math]] \overline{x} = 6/6 = 1, \, \overline{y} = 8.5/6 = 1.41667, \, \hat{\beta}_1 = \frac{\sum x_iy_i - n \overline{x}\overline{y}}{\sum{x_i^2} - n\overline{x}^2} = 0.7. [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.