Hi! So, yours is actually a sophisticated question that masquerades as a simple one, so I'll try to answer this in a way that conveys the concept while perhaps alluding to some of its problems. At its heart, the null hypothesis is a sort of "straw man" that is defined by a researcher at the beginning of an experiment that usually represents a state of affairs that would be expected to occur if the researcher's proposal were false. Note that a null hypothesis is entirely imaginary, and it has nothing to do with the actual state of the world. It is contrived, usually to show that the actual state of the world is inconsistent with the null hypothesis. Suppose a researcher is trying to determine whether the heights of men and women are different. A suitable null hypothesis might be that the difference of the two population averages (height of men and height of women) is equal to zero. Then the researcher would conduct his or her experiment by measuring the heights of many men and women. When it comes time to draw a statistical conclusion, he or she will compute the probability that the observed data (the set of heights) could have come from the null hypothesis (i.e., a world where there is no difference). This probability is called a "p-value". Conceptually, this is similar to a "proof by contradiction," in which we assert that, if the probability is very small that the data could have originated from the null hypothesis, it must not be true. This is what is meant by "rejecting the null hypothesis". It is different from a proof by contradiction because rejecting the null proves nothing, except perhaps that the null is unlikely to be the source of the observed data. It doesn't prove that the true difference is 5 inches, or 1 inch, or anything. Because of this, rejecting the null hypothesis is in NO WAY equivalent to accepting an alternative hypothesis. Usually, in the course of an experiment, we observe a result (such as the observed height difference, perhaps it is ~5 inches) that, once we reject, replaces the hypothesized value of 0 under the null. However, we DON'T know anything about the probability that our observed value is "correct", which is why we never say that we have "accepted" an alternative. I actually hesitate to discuss an "alternative" hypothesis because most researchers never state one and it doesn't matter for the purposes of null hypothesis significance testing (NHST). It is just the name given to the conclusion drawn by the researchers after they have rejected their null hypothesis. Philosophically, there is an adage that data can never be used to prove an assertion, only to disprove one. It includes an analogy about a turkey concluding that he is loved by his human family and is proven wrong upon being slaughtered on Thanksgiving. I'll include a link if I can find it. Now, think about this: The concept of rejecting a null hypothesis probably seems very reasonable as long as we are careful not to overinterpret it, and this is how NHST was performed for decades. But consider - what is the probability that the null hypothesis is true in the first place? In other words, how likely is it that the difference between mens' and womens' heights is equal to zero? I propose that the probability is exactly zero, and if you disagree then I will find a ruler small enough to prove me correct. The difference can never be equal to exactly zero (even though this is the "straw man" that our experiment refutes), so we are effectively testing against a hypothesis that can never be true. Rejecting a hypothesis we already know to be false tells us nothing important ("the data are unlikely to have come from this state that cannot be true"). And since every null hypothesis is imaginary, it is suggested that any null hypothesis can be rejected with enough statistical power (read:sample size). Often a "significant" result says more about a study's sample size than it does about the study's findings, even though the language used in papers/media suggests to readers that the findings are more "important" or "likely to be correct". This has, in part, led to a reproducibility crisis in the sciences and, for some, an undermining of subject-matter-experts' trust in the use of applied statistics. More on reddit.com

I.e the statistic is the same as the parameter. You should start by rethinking this statement. Statistics are variables, calculated from data. Sample statistics are individual realizations of the variable, but they are basically the parameter +/- uncertainty, so we never actually think the statistic is the same as the parameter. why do we include inequalities in the Null Hypothesis This is generally only the case in 1-sided hypothesis testing, not in general. But it basically boils down to how we choose a (1-alpha)% region of the null distribution to define as our critical region / rejection region. If we choose rejection regions from both tails of the null distribution, we have a two-sided hypothesis test, since we are allowing rejections of the null in both directions. If we pull our entire rejection region from only one tail, we maximize our statistical power in that direction, but then we are incapable of rejecting the null at all in the other direction, so we basically incorporate that direction into the null hypothesis statement itself - it's honestly a bastardization of notation, but it does convey the fact that we are only interested in 1 side of the null distribution. I'm not sure at what level you understand these things, so feel free to ask follow-up questions. 2) What purpose does an alternative hypothesis serve? Once we reject the null, the alternative hypothesis is basically the "alternative" to the null that we accept as "likely being true" (maybe a bit of a simplification, but that's the gist). More on reddit.com

You have to think about the epistemology of statistical inference. We are trying to see if we can find "evidence to support a theory". The only way to do that in frequentist stats is to "find evidence inconsistent with another theory". So, if you want to see if "there is evidence to support a positive relationship", you have to see if there is evidence inconsistent with the idea that there is a negative or zero relationship. One of the important factors in digesting this is to realize that if you "fail to reject a null hypothesis", that, in itself, is not evidence that the null is true (many repeated, independent tests with sufficient power that fail to reject will make us "comfortable" with that null...). Regarding equality in the alternate: Since all hypothesis tests must state a specific numerical idea to compare to, you must have an equality statement in the null. The idea behind all hypothesis tests is to try to quantify how inconsistent your data are with the null. If your null has no equality, then which of the uncountably infinite unequal values are you comparing your data to? More on reddit.com

Because not having enough evidence of something doesn't mean that the opposite of that something is necessarily true. More on reddit.com