The first and simplest case of hypothesis testing we will consider is the test of a mean (H0: μ = μ0).

**TEST OF A MEAN (SINGLE SAMPLE, POPULATION VARIANCE KNOWN)**

The first and simplest case of hypothesis testing
we will consider is the test of a mean (*H*_{0}:
μ = μ_{0}). In this case, we will assume that the population
variance is known; thus, we are able to use the *Z* statistic. We perform the following steps for a two-tailed test
(in the next section we will look at both one-tailed and two-tailed tests):

1. State the null hypothesis *H*_{0}: μ = μ_{0} versus the alternative hypothesis *H*_{1} μ: ≠ μ_{ 0}.

2. Choose a significance level α= α_{0} (often we take α_{0} = 0.05 or 0.01).

3. Determine the critical region,
that is, the region of values of *Z* in
the upper and lower α/2 tails of the sampling
distribution for *Z* when μ = μ_{0} (i.e., the sampling distribution when the null
hypothesis is true).

4. Compute the *Z* statistic: *Z* = ( – _{0})/(
σ/√n) for the given sample and sample size *n*.

5. Reject the null hypothesis if
the test statistic *Z* computed in step
4 falls in the rejection region for this test; otherwise, do not reject the
null hypothesis.

As an example, consider the study that used blood
loss data from pigs (refer to Table 8.1). Take μ_{0} = 2200 ml (a plausible amount of blood to lose for a pig in the control
group). In this case, the sensible alternative would be one-sided; we would
assume μ < 2200 for the alternative with the treatment group, because we
expect the treatment to reduce and not to increase blood loss.

However, if we are totally naïve about the
effectiveness of the drug, we might consider the two-sided alternative, namely,
*H*_{1}: μ_{0} ≠ 2200. In this section we are
il-lustrating the two-sided test, so we will look at the two-sided alternative.
We will use the sample data given in Section 8.9 and assume that the standard
deviation σ is known to be 720. The sample mean is 1085.9 and the sample size *n* = 10. We now have enough information
to carry out the test. The five steps are as follows:

1. State the null hypothesis: The
null hypothesis is *H*_{0}: μ = μ_{0} = 2200 versus the alternative hypothesis *H*_{1}: μ_{ }≠_{ }μ_{0 }= 2200.

2. Choose a significance level α= α_{0} = 0.05.

3. Determine the critical region,
that is, the region of values of *Z* in
the upper and lower 0.025 tails of the sampling distribution for *Z* when μ = μ_{0} (i.e., when the null hypothesis is true). For α_{0} = 0.05, the critical values are *Z* = ±1.96 and the critical region
includes all values of *Z* > 1.96 or
*Z* < –1.96.

4. Compute the *Z* statistic: *Z* = ( – μ_{0})/( σ/√n) for
the given sample and sample size *n* =
10. We have the following data: *n* =
10; the sample mean () is
1085.9; σ
= 720; and μ_{0} = 2200. *Z* = (1085.9 –
2200)/(720/√10) = –1114.1/227.684 = –4.893.

5. Since 4.893 (the absolute value of the test
statistic) is clearly larger than 1.96, we reject *H*_{0} at the 5% level; i.e., –4.893 < –1.960. Therefore,
we conclude that the treatment was effective in reducing blood loss, as the
calculated *Z* is negative, implying
that μ < μ_{0}.

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