Level 5: Relationships · Problem 83 of 87

Standard error of the slope

A statistics problem from Statisticsdrill. Solve it with one answer box, with a fresh sample on every attempt and the worked steps and spreadsheet function in the answer, and it is checked the moment you submit.

The task

Data on monthly ad spend and sales at regional stores (x = Ad spend ($k), y = Sales ($k)): (9, 87.8), (10, 98.7), (6, 81.3), (12, 99.1), (15, 111.9), (6, 78.6). Fit the least-squares line and find the standard error of the slope, SE(b) = s_e / √Sxx, where s_e = √(SSE / (n - 2)) is the residual standard error. Round to 3 decimal places.

What this problem teaches

The formula: SE(b) = s_e / √Sxx, s_e = √( SSE / (n - 2) ). Dividing the slope by its standard error gives the t statistic for H0: slope = 0 with n - 2 degrees of freedom. SSE loses two degrees of freedom because the line has two fitted parameters.

A hint if you get stuck

SSE = Σ(y - ŷ)² = Syy - b × Sxy. Then s_e = √(SSE / (n - 2)) and SE(b) = s_e / √Sxx.