# Single-Sample Hypothesis Tests

## Introduction to Single-Sample Hypothesis Tests

• Single-sample hypothesis tests are statistical tests that compare a sample to a population with a known mean, standard deviation and sample size.
• They are used to ascertain if there is a significant difference between the population and sample mean.
• The null hypothesis (H₀) states that there is no significant difference, while the alternate hypothesis (H₁) asserts that there is.

## Types of Single-Sample Hypothesis Tests

• There are two main types of single-sample hypothesis tests: the t-test and the z-test.
• The single-sample t-test is useful when we do not know the population standard deviation. It is more robust against outliers and smaller sample sizes.
• The z-test should be used if we can assume that our data points are independently and identically distributed.

## Running the Tests

• The test statistic is calculated, which tells us how much our sample mean deviates from the null hypothesis population mean.
• If the test statistic is significantly large in size (either positive or negative), this provides evidence that an effect exists, and we can reject the null hypothesis in favour of the alternate hypothesis.

## Interpretation of Results

• The p-value reports the probability of obtaining a result as extreme as, or more extreme than, the result actually obtained, assuming the null hypothesis is true.
• If the p-value is less than our chosen significance level (0.05 is common), we reject the null hypothesis.
• A smaller p-value indicates the effect is robust against sampling variability.

## Assumptions of Single-Sample Hypothesis Tests

• The data must be collected randomly and have independence within the sample.
• The underlying population should be normally distributed if using a t-test. If not, the sample size should be sufficiently large (approximately n > 30) to invoke the Central Limit Theorem for a z-test.
• The variances of the populations must be equal if comparing more than two independent groups (i.e. ANOVA).