The wording I have consistently learned (and taught) has been > we do not have evidence to reject the null hypothesis. That, regardless of what the alternative hypothesis is, we are speaking in terms of rejecting the null hypothesis (or not). The reason for this is that for common statistical tests like t-tests we are proposing a distribution with known parameters as the null hypothesis so that we can test how likely it is that the observations we have made came from that distribution. The alternative hypothesis is often simply "the distribution has different parameterization". In the case of a t-test that is to do with the mean of the distribution, but just simplifying here, More on reddit.com

But they explain it on the website: ... that the hypothesis is correct or that it is probably correct." I doubt if "accept" is the best term in this case as it can lead to confusion. $\endgroup$ ... $\begingroup$ This article here also advocates that the term "accept" should not be used by scientists. $\endgroup$ ... $\begingroup$ "fail to reject the null hypothesis" ... More on stats.stackexchange.com

it's not that we WISH it was different, but that IF it is different, it is an interesting finding. Since we are talking about one sample t-test, let's say you expect mean salary of professional hippo wrestler to be $60k, because idk, most of hippo wrestler you know earns $60k. Then you get the data of wrestler's salaries and run one sample t-test. If mean salary does not differ significantly, you will think "I was right after all" (or to be more strictly speaking, "I can't reject my own claims after all") If mean salary differs significantly, you will think "huh that's interesting. Guess I'm wrong and Ill look into it further" While not exactly true, think null hypothesis like "boring" finding, while alt hypothesis like "cool, interesting, new finding". sometimes, we "like" to fail to reject the null hypothesis. Let's say you personally made 20 lightbulbs with target duration of 48hr. If mean duration of your creation differs significantly from 48 hr, that means you did something wrong while making your lightbulb. You don't WANT to see it, but it's an interesting finding nonetheless. More on reddit.com

Suppose we have a coin, and the null hypothesis is that this coin is fair, it will come up 50% heads and 50% tails on average. Note that this does not mean that it will come up exactly 50% heads and 50% tails in any given sampling. So if we test it with 100 flips and get 52 heads and 48 tails, this corresponds to a fairly high p value. If you flip a genuinely fair coin 100 times, 52 heads and 48 tails is reasonably likely to occur, so we do not reject the null hypothesis. We haven't actually proven that this is a fair coin, it could be a coin which is biased to come up heads 52% of the time and tails 48% of the time, or a coin which is biased to come up heads 55% of the time and tails 45% of the time and happened to get slightly more heads in this sample, but we don't have much proof of that either. The p value was high, which means what actually happened seems likely. Nothing surprising happened, so we have no reason to suspect the coin is unfair. If we flip a coin 100 times and get 81 heads and 19 tails, then this corresponds to a low p value. If you flip a genuinely fair coin 100 times, you are extremely unlikely to get 81 heads and 19 tails, so this is strong evidence that you didn't flip a fair coin, you flipped an unfair coin. We reject the null hypothesis, because the evidence we've observed and the predictions made by the hypothesis do not match very well. Technically it is possible to get 81 heads and 19 tails with a fair coin, it's just extremely unlikely, so the more likely explanation is that you flipped a biased coin that actually gives heads more often. High p values mean the evidence agrees with the null hypothesis, low p values mean the evidence disagrees with the null hypothesis. The higher or lower they are the stronger this agreement or disagreement is. More on reddit.com