# 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.