SECTION 6.5 Hypothesis Testing 399
6.5.1 Hypothesis testing of the mean; known variance
Throughout this and the next section, the null hypothesis will have the
formH 0 : μ=μ 0. However, in the course of rejecting this hypothesis,
we shall considerone-andtwo-sided alternative hypotheses. The
one-sided alternatives have the formHa: μ < μ 0 orHa: μ > μ 0. The
two-sided alternatives have the formHa: μ 6 =μ 0.
A one-sided alternative is appropriate in cases where the null hy-
pothesis is H 0 : μ = μ 0 but that anything ≤ μ 0 is acceptable (or
that anything≥ μ 0 is acceptable). This leads to two possible sets of
hypotheses:
H 0 : μ=μ 0 , Ha: μ < μ 0 ,or
H 0 : μ=μ 0 , Ha: μ > μ 0.Example 1. Suppose that a manufacturer of a mosquito repellant
claims that the product remains effective for (at least) six hours. In
this case, anything greater than or equal to six hours is acceptable and
so the appropriate hypotheses are
H 0 : μ= 6, Ha: μ < 6 ,Therefore, a one-sided alternative is being used here.
Example 2. In the example of precision bolts discussed above, large
deviations on either side of the mean are unacceptable. Therefore, a
two-sided alternative is appropriate:
H 0 : μ=μ 0 , Ha: μ 6 =μ 0 ,Next, one decides on a criterion by whichH 0 is to be rejected; that
is to say, on decides on the probability α of making a Type I error.