Level 4: Inference · Problem 59 of 87

Welch t test: degrees of freedom and decision

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

A plant compares the daily output of two machines. Machine A: n = 20, x̄ = 108, s = 9.7. Machine B: n = 17, x̄ = 112.1, s = 4.4 (units per day). The spreads look unequal, so use the Welch t test for H0: μ1 = μ2 against H1: μ1 ≠ μ2. Find the Welch-Satterthwaite degrees of freedom (do not round them down), then decide at α = 0.05. Round to 2 decimal places.

What this problem teaches

The Welch df is always between the smaller n - 1 and n1 + n2 - 2. Software keeps the fraction; some textbooks round it down, which is slightly conservative.

A hint if you get stuck

With a = s1²/n1 and b = s2²/n2: df = (a + b)² / [a²/(n1 - 1) + b²/(n2 - 1)]. The t statistic is (x̄1 - x̄2) / √(a + b).