STAT 269 - Introductory Statistics
Hypothesis Testing
1. Hypotheses: The first step is to clearly state the hypotheses that we are testing in terms of the
parameters involved, and to clearly define the parameters in terms of the problem at hand. There
will always be two hypotheses, the null and the alternative. For our purposes, the alternative and the
null hypotheses will always be the complements.
•The Null Hypothesis will always contain a statement of equality. Generally this is the “status
quo” hypothesis, that nothing has changed from some standard, or that the subset of interest has
the same characteristic as the larger population.
•The Alternative Hypothesis is often called the “research hypothesis” and is generally a statement
of what we believe, or hope, to be true.
2. Assumptions: Each method we discuss will make certain assumptions about the way we collected
our data and the nature of the population, or populations, from which they came. Stating these
assumptions with each test will remind us what we are assuming, and that these assumptions can
affect the validity of the test. If any of the assumptions are not valid, the conclusions we reach may
not be trustworthy.
3. Rejection Region: In this step we will determine what values of the Test Statistic (see the next
step) will lead us to reject the null hypothesis in favor of the alternative, and which values will not
enable us to reject the null hypothesis.
4. Test Statistic: This is a statistic (based on our data) whose distribution is known under the as-
sumption that the null hypothesis is true. If the value is abnormal for this distribution, it causes us
to doubt, and possibly reject, the null hypothesis.
5. P-value: This is a measure of the probability, assuming the null is true, that we would see a test
statistic as unusual, or “rare”, as the one we observed.
6. Conclusion: We will then compare the Test Statistic to our Rejection Region, and our P-value to our
predetermined cut off value αand determine our conclusion to either reject the null hypothesis or fail
to reject the null hypothesis. Note that we will never “accept the null hypothesis”. This conclusion
should be stated clearly in terms of the problem at hand, in plain English. “Reject the null” or “Fail
to reject the null” is not sufficient.
Hypothesis Testing Details: For µ, large sample or σknown
1. Hypotheses:
H0:µ=µ0µ≤µ0µ≥µ0
Ha:µ6=µ0µ>µ0µ<µ0
Where: µis the mean for all
2. Assumptions: We have independent, random observations from some population, and the sample
size is large enough that we can use the Central Limit Theorem.
