# Using the normaldistribution in hypothesis tests

## Using the normaldistribution in hypothesis tests

# Using the Normal Distribution in Hypothesis Tests

## Introduction to Hypothesis Testing with Normal Distribution

**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.

## Constructing a Hypothesis Test

- 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.

## Making Decision and Interpretation

- 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.

## Type I and Type II Errors

- 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.