CK-12 Probability and Statistics - Advanced

8.1. Hypothesis Testing and the P-Value http://www.ck12.org

H 0 :μ= 250

Ha:μ= 253

By specifying a value for the alternative hypothesis, we have selected one of the many values forHa. In determining
the power of the test, we must assume thatHais true and determine whether we would correctly reject the null
hypothesis. In other words, we want to determine the power of our test for detecting this difference. In this example,
we may choose a certain dosage if there were medical repercussions above that level.

We want to find the area under the curve that is associated with making a Type II error. In the example above, this
means that we need to find the power that the test has for detecting this difference. Calculating the exact value for
the power of the test requires determining the area above the critical value set up to test the null hypothesis when
it is re-centered around the alternative hypothesis. Say that we have an alpha level of.05 – we would then have a
critical value of 1.64 for the single-tailed test which would have a value of:

z=

X ̄−μ

σX

1. 64 =

X ̄− 250

50 /

√

200

X ̄= 1. 64

(

50

√

200

)

+ 250 ≈ 255. 8

Now, with a new mean set at the alternative hypothesis(Ha:μ= 253 )we want to find the value of the critical score
( 255. 8 )when centered around this score. Therefore, we can figure that:

z=

X ̄−μ

σX

=

255. 8 − 253

3. 54

≈ 0. 79

Using the standardzdistribution we find that the area to the right of az-score of.79 is. 2148 .This means that since
we assumed the alternative hypothesis to be true, there is only a 21.5% chance of rejecting the null hypothesis. The
power of this test is about 0. 215 .In other words, this test of the null hypothesis is not very powerful and has only a
0 .215 probability of detecting the real difference between the means.

There are several things that affect the power of a test including:

• Whether the alternative hypothesis is a single-tailed or two-tailed test.


• The level of significance(α).


• The sample size.

Lesson Summary

1. Hypothesis testing involves making educated guesses about a population based on a sample drawn from the
   population. We generate null and alternative hypotheses based on the mean of the population to test these
   guesses.


2. We establish critical regions based on level of significance or alpha(α)levels. If the value of the test statistic
   falls in these critical regions, we are able to reject it.