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I have this multi-part question on an assignment that I don't understand. Hopefully someone can help.

There's a story that 4 students missed their final and asked their professor for a make-up exam claiming a flat tire as their excuse. The professor agreed and put them in separate rooms. The first question of the test was easy and only worth a few points. The second question comprised all of the remaining points and asked "which tire went flat? RF, RR, LF, LR?"

  1. Which hypothesis set is true? HS1: Ho: Students told the truth. HA: Students lied OR HS2: Ho: Students lied. HA: Students told the truth

I think that HS2 is the correct answer but I'm not certain as I feel like it could be either depending on how you look at it.

  1. What is the rejection region of your test?

I know this relies on question 1 but I'm not sure how to move forward in general not to mention that I'm not positive of my answer to question 1.

  1. What is the Type 1 error rate alpha of the rejection region you defined?

  2. After collecting the students answers, how do you define the p-value of such answers for the test?

I'm really so confused about all of this. So I'm really hoping someone can help.

Courtney
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  • I guess the four students were all in the same car? Regarding which null hypothesis to use...normally, we prefer to assume "innocent until proven guilty". – Brad S. Mar 18 '14 at 16:15

2 Answers2

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Hm ... how about this:

H0: flat tire HA: students lied

If H0 holds then the probablity of different answers is exactly 0 (assuming students are in a good memory).

1) Hence, reject H0 if there are any different answers and the p-value is exactly 0. 2) Keep H0 if answers agree and the power of the test is 1-1/256.

user230195
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After giving this some thought, I think that the professor assumes that the students are lying - why else would he concoct such a scheme to have the students prove otherwise?

$ \mathbb{H}_0:$ The students did not have a flat tire.
$ \mathbb{H}_a:$ The students did have a flat tire.

Our test statistic will be the number of answers that agree. The decision criteria is that if all four answers are the same, then he rejects the null hypothesis and accepts the alternative. That means the rejection region is N < 4, where N is the observed number of answers that agree.

The idea is that if the null hypothesis is true and they did not have a flat tire (and they were not smart enough to make up and agree upon a detailed story), then it is unlikely that all four answers to the question "which tire was flat?" will agree.

Now, under the null hypothesis, no tire went flat so, all four students will have to guess which tire went flat. Each one randomly says one of the four and their answers are all independent of the others. There are $4 \times 4\times 4\times 4 = 256$ possible different ways the students can answer but only four of those 256 have all four agreeing on which tire blew out. So, the probability that all four answers will agree (under $\mathbb{H}_0$) is 4/256 or about 0.015625. That means that the probability of rejecting the null hypothesis when it is true (and committing a type I error) is $\alpha = 0.015625$

Brad S.
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