# Using the norma ldistribution in hypothesis tests

## Using the norma ldistribution in hypothesis tests

## The Normal Distribution and Hypothesis Testing

- Hypothesis testing involves making an initial assumption, or
**null hypothesis (H0)**, about a population parameter. - The
**alternate hypothesis (H1)**is what you might believe if the null hypothesis is deemed unlikely. - The normal distribution plays a key role in hypothesis testing due to the
**central limit theorem**, which states that the sum of a large number of independent and identically distributed random variables tends towards a normal distribution, regardless of the shape of the original distribution.

## Using Z-Scores in Hypothesis Testing

- A
**Z-Score**refers to how many standard deviations an element is from the mean. - For hypothesis testing, the Z-Score helps us decide whether to accept or reject the null hypothesis.
- Under the null hypothesis, we assume that our test statistic follows a standard normal distribution. Then, we calculate the Z-Score of our observed statistic.
- If our calculated Z-Score is more extreme than the critical value, we reject the null hypothesis in favour of the alternative. This is often known as the
**Z-Test**.

## Type I and Type II Errors

- A
**Type I error**occurs when we incorrectly reject the null hypothesis. This is also called a false positive. - A
**Type II error**occurs when we fail to reject the null hypothesis when it is actually false. This is also known as a false negative. - The probabilities of making Type I and Type II errors are denoted as
**alpha (α)**and**beta (β)**, respectively.

## Confidence Intervals and Hypothesis Testing

- Hypothesis tests can also be performed using confidence intervals.
- A
**confidence interval**is a range of values, derived from a data set, that is likely to contain the value of an unknown population parameter. - If a certain hypothesized value is not within the confidence interval, we can reject the null hypothesis at the corresponding significance level.
- Generally, the confidence level complements the significance level (a 95% confidence level corresponds to a 5% significance level).

## Applications

- Hypothesis testing using the normal distribution is a fundamental technique in statistics and it’s widely used in fields such as social sciences, finance, and engineering for making inferences and decisions under uncertainty.
- It’s essential to consider the balance between Type I and Type II errors as this can significantly affect the results and conclusions of a hypothesis test.