Power function for a test that a normal population has mean zero versus a two-sided alternative

**TWO-SAMPLE t TEST (INDEPENDENT
SAMPLES WITH A COMMON VARIANCE)**

Recall from Section 8.5 the use of the appropriate *t* statistic for a confidence interval
under the following circumstances: the parent populations have normal distribu*s*2*p*, calculated by the formula *Sp*2 = {*St*2(*nt* – 1) + *Sc*2(*nc* – 1)}/[*nt* + *nc *– 2]

**Figure 9.2. **Power function for a test that a
normal population has mean zero versus a two-sided alternative when the sample
size *n* = 25, *n* = 100, and the significance level α = 0.05.

Suppose we want to evaluate whether the means of
two independent samples selected from two parent populations are significantly
different. We will use a *t* test with *s _{p}*

1. State the null hypothesis *H*_{0}: μ* _{t}* = μ

≠
μ* _{c}*.

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 *t* in
the upper and lower α/2 tails of the sampling
distribution for Student’s *t*
distribution with *n _{t}* +

4. Compute the *t* statistic: for the given sample
and sample sizes *n _{t}* and

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

We will apply these steps to the pig blood loss
data from Section 8.7, Table 8.1. Recall that *S _{p}*

1. State the null hypothesis *H*_{0}: μ* _{t}* = μ

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

3. Determine the critical region,
that is, the region of values of *t* in
the upper and lower 0.025 tails of the sampling distribution for Student’s *t* distribution with 18 degrees of
freedom when μ* _{t}*/μ

4. Compute the *t*
statistic: We are given that the sample sizes are *n _{t}*
= 10 and

5. Now, since –2.362 < –*C* = –2.101, we reject *H*_{0}.

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