A professor gives a multiple choice exam where each question has four choices. The professor decides not to count a question against students if the class as a whole does worse on the question than they would have done simply by guessing. That is, if significantly less than 25% of the students answer a question correctly, then that question won't count against them.
Let p represent the proportion of students that would correctly answer the question.
Which of the following is an appropriate set of hypotheses for such a significance test?

ANSWER: H0:p=0.25
Ha:p<0.25

There are two hypotheses. First one is called null hypothesis and it is chosen such that it predicts nullity or no change in a thing. It is usually the hypothesis against which we do the test. The hypothesis which we put against null hypothesis is alternate hypothesis.

Null hypothesis is the one which researchers try to disprove.

For the given case, the test is being done to check if there is significance or not.

The null hypothesis would assume that the score is not significantly less than 25%, as null hypothesis stays with nullity or denial of what is being prominently tested, which is the fact that students performed significantly less than 25%

Thus, we get null hypothesis as: [tex]H_0 : p \geq 25\% = 0.25[/tex] (not significantly less than 0.25, so either proportion is 0.25 or bigger than 0.25)

Alternate hypothesis is the hypothesis statement we want to test, which is the statement that students performed significantly less than 25%.

or

Alternate hypothesis: [tex]H_1 : p < 0.25[/tex]

Thus, the pair of hypotheses for the given situation is given as: [tex]H_0: p \geq 0.25\\H_1: p < 0.25[/tex]

Learn more about null and alternate hypothesis here:

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