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Constructing Hypotheses

Once you've decided that a hypothesis test is the appropriate direction to go, you will need to construct your two hypotheses. Initially (in our course), our hypotheses will be statements about either µ (a population mean) or p (a population proportion). You should be able to tell from the wording of the problem if you are testing about some kind of average/mean or some kind of percentage/proportion.

Example 1: Test to see if more than 60% of voters prefer Candidate X.

Example 2: Test to see if the average age in your dorm is 19, or if it is lower than that.

On Example 1, we can see that the question involves a percentage of people, and thus we use the parameter p. On Example 2, the question involves the average of something, so the parameter used would be µ.

Next we must decide what the problem actually wishes to know about the parameter. It might say one thing about the parameter, or it might say two things. We must write those statements in mathematical form, using either µ or p, and using one of <, >, ≤, ≥, = or ≠.

On Example 1, it says "more than 60% of ." We've established that we are using the parameter p, and this statement says we want our parameter to be more than 60%. That means we must use "p > 0.60" as one of our hypotheses. There is nothing in the statement that tells us directly what the other hypothesis should be, so we must choose the opposite of the first hypothesis as our second hypothesis. The opposite of "p > 0.60" would be "p ≤ 0.60," so that is our second hypothesis.

On Example 2, it says "average age is 19, or it is lower than that," which is saying two different things. We know our parameter is µ, and the first phrase tells us that µ should be 19, so one hypothesis must be "µ = 19." The second phrase tells us that µ should be lower than 19, so we use "µ < 19."

Lastly, we must decide which hypothesis should be H_{0} and which should be H_{1}. One of our two hypotheses must include an equal sign, meaning it must use either =, ≤ or ≥. (The signs ≠, < and > do not count as "including an equal sign.") The hypothesis that does include the equal sign becomes H_{0}, while the other hypothesis becomes H_{1}.

On Example 1, our hypotheses are "p > 0.60" and "p ≤ 0.60." So we identify: H_{0}: p ≤ 0.60, and H_{1}: p > 0.60.

On Example 2, our hypotheses are "µ = 19" and "µ < 19." So we identify: H_{0}: µ = 19, and H_{1}: µ < 19.

Example 3: Test the claim that the average age of ENC professors is 52. Our parameter is µ, and the problem states that "µ = 52" must be one of our hypotheses. No other hypothesis is stated, so we take "µ ≠ 52" as our other hypothesis. Thus, H_{0}: µ = 52, and H_{1}: µ ≠ 52.