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The concepts are very useful in biological, economical and social experiments and all kinds of generalizations based on information about a smaller subset.For example, if an experimenter takes a survey of a group of 100 people and decides the presidential votes based on this data, the results are likely to be highly erroneous because the population size is huge compared to the sample size.
In the study of statistics, a statistically significant result (or one with statistical significance) in a hypothesis test is achieved when the p-value is less than the defined significance level.
The p-value is the probability of obtaining a test statistic or sample result as extreme as or more extreme than the one observed in the study whereas the significance level or alpha tells a researcher how extreme results must be in order to reject the null hypothesis.
If you want to generalize the findings of your research on a small sample to a whole population, your sample size should at least be of a size that could meet the significance level, given the expected effects.
Expected effects are often worked out from pilot studies, common sense-thinking or by comparing similar experiments. Comparing the statistical significance and sample size is done to be able to extend the results obtained for the given sample to the whole population.
This is to be expected because larger the sample size, the more accurately it is expected to mirror the behavior of the whole group.
Therefore if you want to reject your null hypothesis, then you should make sure your sample size is at least equal to the sample size needed for the statistical significance chosen and expected effects.
This means that the survey needs higher power to accept a hypothesis.
Some researchers choose to increase their sample size if they have an effect which is almost within significance level.