There are several conceptual errors in what you wrote you had been instructed.

Very generally speaking, you do hypothesis tests to find out if your observed data contains enough information for some kind of interpretation. That's all. We call it a "(statistically) significant result" when there is enough information for an interpretation, and otherwise we call it a "(statistically) non-significant reesult". This only means that the observed data don't give us enough information of the the feature we like to interpret.

The previous paragraph was very general, but it contains an extremely important message: why, in general, we do tests at all and what the test results tell us (and what they don't tell us). About 80% (at least) of all applied stats books I know get this wrong, and so do most researchers.

This principle idea will become clearer with a more concrete example. A prototype example is the effect of a treatment on some quantitative response variable. In your statistical model you formulate the response variable as a random variable of some distribution family. The interesting feature of this random variable is its expected value which may depend on the treatment. We don't know the expected value of this distribution. However, obsered data gives us some information about this value, which I will denote with "µ": for any possible expected value of µ we can use the distribution model to calculate how likely the observed data would be, if µ were what we set it to be. We can so get the likelihood of the data for any hypothesized value of µ. The likelihood will be larger when we choos a µ somewhere in the "middle" of the observed values, and it will decrease the further away from the data we coose a value of µ.

The value of µ where the likelihood is maximal is called the maximum likelihood estimate (of µ). This is the sample mean which I will denote with "m".

If we have a lot of observations, this drop will be steep, so that the likelihood will be much lower we we choose a value that is just slightly different to the sample mean. If we have only few observations, the likelihood won't change so quickly: there is a larger interval of value we can choose for µ and still find a relatively large likelihood of the data. This shows that and how data bring information about µ.

You may now be interested in finding out if that information about µ is sufficient to expect µ being on the same side of some value h like the sample mean m. If the difference between m and h is small and the sample size is small, the ratio of likelihoods for m and h will be close to 1. But when there is a lot of data, the likelihood drops quickly around m and the likelihood ratio will be much smaller than 1.

The genious trick is that one can derive a probability distribution of this likelihood ratio under the assumption that µ = h. So it is possible to calculate the probability of even smaller likelihood ratios than the one we obtained for the observed data. This is the famous p-value. If this probability is very small, then the observed data are very incompatible with the statistical model and our hypothesis that µ = h. Therefore, the data is even more incompatible with all values for µ that are on the opposite side of h than m (because there the likelihood ratios will be even smaller). When, for instance, m > h, and the data are very incompatible with the assumption that µ = h, then the data are even more incompatible with any µ < h. But this means: when there is any value compatible with the data, then this value must be larger than h. This means we "reject the hypothesis that µ < h" and conclude that µ > h (because the data is too incompatible with any µ < h).

Often, it is stated that only µ = h is rejected.Although this is not wrong, it obscures that therewith all hypotheses µ < h or µ > h are also rejected, depending on whether m < µ or m > µ.

Regarding your "Bayesian bypass": this adresses an entirely different question. In the hypothesis test you try to find out if the information in the data is sufficient for a statement about the sign of the difference h-m. The Bayesian analysis requires a prior distribution for µ and modifies this based on the information in the data to a posterior, expressing your current knowledge about µ (and notif you have a large enough sample size to make a statement about whether µ > h or µ < h).

The problem you have encountered with hypothesis testing is the problem of conflating a scientific hypothesis with a statistical hypothesis. Statistical hypothesis testing does not produce practical results. Statistical testing concerns the achievement of a level of data quality.

The conventional formation of a statistical hypothesis is to state 'there is no difference,' but, it is equal to state 'there is a difference.' The latter statement invites the question 'How much is needed to show a difference?' There is an important distinction between 'show a difference' and 'show a practical difference.'

Statistical texts rarely use the approach of 'How much is needed to show a difference.' The usual approach is to use the null hypothesis and achieve statistical significance. There is no difference in outcome for either approach. There is a difference in concept. The null hypothesis concept ignores the scientific need to show a practical difference.

The statistical null hypothesis leads to the confusion that a statistical hypothesis is a scientific hypothesis. The word hypothesis can be dropped from a statistical discussion with no loss of meaning. Null is more concise than null-hypothesis. Statistical testing does not need the addition of the word hypothesis. The word hypothesis deflects the meaning away from evaluating data quality.

The question 'How much is needed to show a difference?' is a data quality question. The term 'statistically different' based on a p-value shows an acceptable difference at a predetermined alpha. The data at statistical significance is the minimum necessary to show a possible difference. See the answer of Jochen Wilhelm .

Statistical significance is a data quality level that shows a difference, but that level does not allow quantification of the difference. It is not scientifically practical. Jochen Wilhelm indicated that the observed data contain enough information to allow some interpretation. All interpretations are limited to data interpretations.

Hi Deepthi Balla ,

In addition to the valuable answers of Jochen Wilhelm and Joseph L Alvarez ,

two papers (one old but by a giant statistician):

Statistical Significance and Biological Relevance

https://efsa.onlinelibrary.wiley.com/doi/epdf/10.2903/j.efsa.2011.2372

Lady tasting tea

https://web.ma.utexas.edu/users/mcudina/m358k-lady-tasting-tea-original.pdf