Statistical Inference: A Short Course

10.8 Hypothesis Tests for μ (σ Unknown And n Small)

Given that σ is unknown and must be estimated by s and the sample size n is small, we must use the t distribution. To this end, let us consider the following tests.

Example 10.4

A manufacturer of security lamps claims that, on the average and under normal conditions, they will not draw more than 2.20 amps of current. A sample of 16 security lamps yielded = 2.65 amps with s = 0.77 amps. Should we reject the manufacturer's claim at the α = 0.05 level? Here we seek to test H₀: μ = μ_o = 2.20, against H₁: μ > 2.20. Then

With , we see that . Since t_o falls within the critical region, we reject H₀ (the manufacturer's claim) at the 0.05 level of significance.

Have we proved the manufacturer wrong? The answer is “no.” It is important to remember that in hypothesis testing we are never “proving” or “disproving” anything—we are simply making an “assumption” H₀ about μ, gathering some data, and then, in terms of a given probability level α, determining if we should reject that assumption on the basis of sample evidence. Nothing has been proven. ...