Hypothesis testing with normal distribution

[Cheman]

Hypothesis testing with normal distribution...

I've been learning about Hypothesis testing with normal distribution, but I don't understand the need for the significance level. By this I mean that i understand that according to the Central Limit Theorem a distribution of the means will be a normal distribution (for a sufficiently large value of n ) with a mean of the mean of the initial population and a standard deviation equal to the standard deviation of the population divided by the square root of n. However, to see whether we think a sample no longer fits the original popultion (as is the aim of hypothesis testing) I would have initially guessed that you would see if the mean of the sample being tested was an outlier - ie: its mean was 2 standard deviations out of the "mean distribution". However, this is apparently not the case - we instead use a significance level, and this should apparently tell us the boundary for whether the teasted sample is still relevant to the parent population or not - even if this significance level is above or below 2 sds of the mean; its is this significance level that matters not whether the mean of the tested sample is an outlier. Why is this the case?

Thanks in advance.

 

[BicycleTree]

With hypothesis testing you can determine what the actual probability of making an error is. Saying "reject the hypothesis if the sample statistic is 2 st. dev. away from it" is just an arbitrary rule-of-thumb.

 

[tacman]

The significance level is an essential component in hypothesis testing with normal distribution because it helps us determine the probability of rejecting the null hypothesis when it is actually true. In other words, it tells us how confident we can be in our conclusion from the sample data.

If we were to solely rely on the mean of the sample being an outlier, we would not have a clear threshold for determining whether the sample is still representative of the population or not. The significance level, on the other hand, provides us with a specific cutoff point for determining if the results of our sample are statistically significant or not.

Additionally, the significance level takes into account both the mean and the standard deviation of the sample, providing a more comprehensive understanding of the data. It helps us account for any variations in the data and ensures that our conclusion is not solely based on one outlier.

In conclusion, the significance level is a crucial aspect of hypothesis testing with normal distribution as it helps us make a more informed and accurate conclusion about the relationship between the sample and the population. It takes into account both the mean and standard deviation, providing a more comprehensive understanding of the data and reducing the chances of making an incorrect conclusion based on outliers.