Question about the logic behind hypothesis testing

Let us say that we have this following problem:

"A government agency claims that more than 50% of US tax returns were filed electronically last year. A random sample of 150 tax returns for last year contained 86 that were filed electronically. Test the claim at $\alpha = 0.05$ significance level."

In this problem our null hypothesis is: $H_0 : p \leq 0.50$ and $H_a : p > 0.50$. I've calculated the sample proportion : $\frac{x}{n} = \frac{86}{150} = 0.573$ and the standard error: $\sqrt{\frac{p(1-p)}{n}} = 0.0408$. Now my z-score is 1.79 which is greater than $z=1.64$, so I'm "required to reject the null hypothesis." But my question is how do I know that 86 out of 150 is not an unusual sampling to begin with? Why can't we reject the random sampling and support the claim given by $H_0$?

My last question deals with what to do with $H_a$. Once $H_0$ has been rejected do we say that there is "sufficient evidence for $H_a$" or do we say that "$H_a$ is true but under these conditions :"?

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2 Answers

how do I know that 86 out of 150 is not an unusual sampling to begin with?

You don't know that 86 out of 150 is not an unusual sampling. It may be that in the population only, say, 45% of people file their returns electronically, and you've just been unlucky and found an unusual sample that doesn't reflect this property of the underlying population.

Samples are random, and hence so is the result of a hypothesis test is random (before you carry it out). A practical interpretation of the 0.05 significance level is that 5% of the time you will choose a sample that wrongly rejects $H_0$: this is known as type 1 error. Note also that confidence intervals are very strongly related to hypothesis tests and similar comments apply.

My last question deals with what to do with $H_a$. Once $H_0$ has been rejected do we say that there is "sufficient evidence for $H_a$" or do we say that "$H_a$ is true but under these conditions :"?

Related to your previous question, you should say something like "We reject $H_0$ at the 5% significance level, in favour of $H_a$". I.e., you're qualifying the rejection with how likely you think you've rejected erroneously.

Assuming that you're studying for some kind of academic qualification you would be well advised to memorize their own particular form of words.

According to how the exercise is set up I would have tested $H_0: p=0.5$ against $H_a : p<0.5$ since the agency "claims" that more than 50$\%$ were filed electronically, so $H_0$ should usually be taken as what it is believed or what one wants to either refute or reject.

On the other hand, one has to "assume" or expect that no "chance " or "fortuitousness" has happened during the sampling. I mean, one should believe that the sample is accurate enough to trust on it. Of course, there is possibility for an error of type I which means you reject $H_0$ when it is in fact true, and this might be because of a "bad" or non-representative sample as you mention.

It is very important that in hypothesis testing one is not proving anything. One is just trying to assess evidence against some hypothesis, that is, you try to find evidence in a data set in order to REJECT a hypothesis, but never in order to ensure prove something. In this case data does not give potential evidence that more than 50$\%$ of US tax Returns were filed electronically.

Finally, one should compute the so-called $P$-vaue of the test, which gives an idea of how much or how little evidence there is for $H_0$ to be true.