# Hypothesis testing with normal distribution

• Cheman
In summary, hypothesis testing with normal distribution involves using a significance level to determine whether a sample is still relevant to the parent population, instead of just looking at whether the sample mean is an outlier. This allows for a more accurate determination of the probability of making an error.
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?

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.

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.

## 1. What is a hypothesis test with normal distribution?

A hypothesis test with normal distribution is a statistical method used to determine whether a sample from a population is likely to have come from a specific distribution, such as a normal distribution. It involves comparing the observed data to the expected data based on a null hypothesis and determining the likelihood of obtaining the observed data if the null hypothesis is true.

## 2. What is a null hypothesis in hypothesis testing with normal distribution?

The null hypothesis in hypothesis testing with normal distribution is a statement of no difference or no effect. It assumes that any observed differences or effects in the data are due to chance and not a result of the variables being tested. It is typically denoted as H0.

## 3. How is the p-value used in hypothesis testing with normal distribution?

The p-value in hypothesis testing with normal distribution is the probability of obtaining the observed data or more extreme data if the null hypothesis is true. It is used to determine the level of statistical significance, with a lower p-value indicating stronger evidence against the null hypothesis. Typically, a p-value of 0.05 or less is considered statistically significant.

## 4. What is the difference between a one-tailed and two-tailed test in hypothesis testing with normal distribution?

A one-tailed test in hypothesis testing with normal distribution only considers one direction of the distribution, either the left or right tail. It is used when the researcher has a specific hypothesis about the direction of the effect. In contrast, a two-tailed test considers both directions of the distribution and is used when the researcher does not have a specific hypothesis about the direction of the effect.

## 5. How is the confidence level related to hypothesis testing with normal distribution?

The confidence level in hypothesis testing with normal distribution is the probability that the true population parameter falls within a certain range of values. It is typically set at 95%, meaning that there is a 95% chance that the true population parameter falls within the confidence interval. This is related to hypothesis testing as the confidence interval can be used to determine the level of statistical significance and support for the null hypothesis.

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