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Hello Parkerha,

we learned only binomial and normal distribution so I am not sure what a t and z test is, but I think you are talking about statistical hypothesis test, right?

I think I can only try to answer your last question: in most tests is as a hypotesis a p value given (as ist should be), for example: A woman claims that she can tell water from the bottle and the tap apart. Our hypotesis H_0 is p = 0.5, i.e. the probability that she will guess right is 50%. Our alternative hypothesis is H_1 is p > 0.5, i.e. she can really distinguish the water by its taste.

But if we try ten times and she guesses six times right, it doesn't mean that she really has the skill, for the probability to guess randomly at least 6 from 10  times right is still 0,62! So to decide whether she has the skill or not you (or your teacher who makes the exercise) have to define a significance level, which is most often round about 2%. This means: If the woman guesses right so often, that if she were only guessing, the probability to come to this result would only be 2% or less, then we will believe her.

To come back to your question: If we have a both-sided test we have to divide the value of the significance niveau. For example: If we look wether a coin is fair, we must make a both-sided test, for the coin can fall with a higher probability on the one side or on the other. If our significance niveau is 2% again, we have now 1% for the one side and 1% for the other side. Which means: only if it falls so often on one side, that the probability to get this result with a fair coin is less then one percent, we assume, that the coin is manipulated. This applies to both sides.