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.
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 = μo = 2.20, against H1: μ > 2.20. Then
With , we see that . Since to 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. ...