8.1. Hypothesis Testing and the P-Value http://www.ck12.org
Solution:
First, we develop our null and alternative hypotheses:
H 0 :μ= 145Let’s calculate the test statistic for several different scenarios.
Example:
Ha:μ> 145This alternative hypothesis is>since we are only concerned with the podgainwhich translates to above the mean.
Next, we calculate the standardz-score for the sample of pea plants.
z=
X ̄−μ
σX=
147 − 145
100 /
√
144
= 0. 24
In the following lessons, we will use these standardz-scores and the critical regions to evaluate the null and the
alternative hypotheses.
Testing the P-Value of an Event
We can also evaluate a hypothesis by testing the probability, or the P-value, of an event occurring. When we assume
that we have normal distributions, we can determine approximately where on the normal distribution that the sample
mean will fall. When we know where it falls, we can determine theprobabilityof obtaining a sample value either
greater or smaller than the mean by using thez-score.
Let’s use the example about the pea farmer. As we mentioned, the farmer is wondering if the number of pea pods
per plant has gone up with his new planting technique and finds that out of a sample of 144 peas there is an average
number of 147 pods per plant (compared to a previous average of 145 pods). But the farmer is really hoping that
some plants have a more dramatic yield increase. What is the probability of a plant having a much higher yield of
over 155 pea pods?
To find this probability, first find thez-score for the hypothesized sample mean using the formula that we learned in
the section above. Therefore, az-score for a sample of plants with 155 pods would be:
z=X ̄−μ
σX=
155 − 145
100 /
√
144
= 1. 20
Using thez-score distribution, we find that the area beyond az-score of 1.20 is equal to. 1151 .This means that there
is.1151 or 11.5% chance that a pea plant will produce over 155 pods.
Type I and Type II Errors
When we decide to reject or not reject the null hypothesis, we have four possible scenarios: