Using the normaldistribution in hypothesis tests

Using the Normal Distribution in Hypothesis Tests

Hypothesis testing is a statistical method that uses sample data to evaluate a claim about the parameters of a population.
Hypothesis testing often utilizes the normal distribution, especially when the size of the sample is large (n > 30), or the population is known to be normally distributed.
In hypothesis testing, we make two conflicting assumptions about the population: the null hypothesis (H₀) which is the assumption to be tested, and the alternative hypothesis (H₁) which is what we consider as evidence against the null hypothesis.

Begin by identifying a null hypothesis (H₀) and an alternative hypothesis (H₁).
Determine a significance level (α), which is the maximum probability you are willing to accept of rejecting the null hypothesis when it is, in reality, true. A common choice for α is 0.05.
Based on the sample data and the identified significance level, calculate the test statistic, which can be standardised to follow a standard normal distribution (z-distribution).
If you have a large sample size (n > 30) or if the population standard deviation is known, use the z-test. Compute the test statistic z = (x̄ - μ₀) / (σ/√n), where x̄ is the sample mean, μ₀ is the population mean under the null hypothesis, σ is the population standard deviation, and n is the sample size.
For smaller samples from a normally distributed yet unknown standard deviation population, the t-test is used instead of the z-test.

Compare the test statistic to the critical value corresponding to the significance level. The critical value can be found from standard normal distribution tables.
If the test statistic is more extreme in the direction of the alternative hypothesis than the critical value, we reject the null hypothesis (H₀) and support for the alternative hypothesis (H₁).
If the null hypothesis is rejected, the result is said to be statistically significant.
It’s important not to confuse statistical significance with practical significance. Even a small effect can be statistically significant with large enough sample sizes.
The choice of the significance level is somewhat arbitrary, and should reflect the consequences of making a Type I error - rejecting the null hypothesis when it is true.

A Type I error occurs when the null hypothesis H₀ is true, but is rejected. The probability of making a Type I error is equal to the significance level α.
A Type II error happens when the null hypothesis is false, but is not rejected. The probability of making a Type II error is denoted by β. The power of a test (1 - β) is the probability that it correctly rejects a false null hypothesis.
The risks of Type I and Type II errors should be balanced, based on the potential consequences of these errors in the given context.