Statistics: confusion regarding type I and type II errors.

Question

My textbook has the following question:

Here is how I approached this problem:

If the allergist wishes to test the hypothesis that at least 30% of the public is allergic to some cheese products, then the following are true:

$H_0: p<0.30$

$H_1: p\geq0.30$

(a) A type I error occurs when we reject the null hypothesis ($H_0$) when it was in fact true. So this would occur if $p < 0.30$ but we concluded that $p\geq 0.30$.

(b) A type II error occurs when we fail to reject the null hypothesis when there was sufficient evidence suggesting it is false. So this happens if $p\geq 0.30$ but we conclude that $p<0.30$

Here's what the book's answer key says, though:

This seems to be the exact opposite...and the book does the same for all other problems. Am I misunderstanding something about Type I and II errors?

Answer 1

Generally the null hypothesis includes the condition of equality. So I imagine that the book set up its null and alternative hypotheses the opposite of how you did.

Answer 2

There may be some confusion on how to interpret nulll hypothesis. In this case where a hypothesis itself is being tested, the allergist was testing the hypothesis of p>0.03 as null hypothesis.

There is another interpretation of null hypothesis in the context of significance testing. Here, the true null hypothesis means the sample statistics is not significant enough to reject the null hypothesis. So null hypothesis would be when there is no significant allergic reaction.

Since we are not testing significance (such as whether certain cheese products are allergic to the public or not), but only testing the hypothesis about the allergic population probability, the first interpretation should be used.

Answer 3

Your set up of the hypothesis test is absolutely correct.

But, in your textbook you see a typical case of "applied statistics" where the so called "working hypothesis" ($\geq 30\%$ allergic) - the claim - is set to be $H_0$.

Unfortunately your textbook messes around with the concept of "hypothesis". It is not clear from the exercise text, what $H_0$ is. You guessed logically correctly that the claim to be tested is most probably "$\geq 30\%$ allergic". But the textbook did exactly the opposite without indicating any reason for that.

Do not let you lead astray by above statements that the "="-sign has to be part of $H_0$. This is simply not true, although some "educational" or "applied" texts may state this. It just happens often to be that way.

Only a short note on your (b): "... we fail to reject the null hypothesis when there was sufficient evidence suggesting it is false." The sample statistic did NOT provide sufficient evidence suggesting $H_0$ might be false.