Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population parameter based on sample data. It involves setting up two competing hypotheses: the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis represents a statement of no effect or no change, while the alternative hypothesis represents what we are trying to find evidence for. We calculate a test statistic from our sample data and compare it to a critical value or use it to find a p-value. The significance level (α) is the probability of incorrectly rejecting the null hypothesis when it is true (Type I error). A one-tailed test is used when the alternative hypothesis specifies a direction (e.g., greater than or less than), while a two-tailed test is used when the alternative hypothesis specifies a difference in either direction (e.g., not equal to). The critical region is the range of values for the test statistic that would lead to the rejection of H₀. The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming H₀ is true. If p-value < α, we reject H₀.

A Type I error occurs when the null hypothesis is rejected when it is actually true. The probability of a Type I error is equal to the significance level, α. A Type II error occurs when the null hypothesis is not rejected when it is actually false. The probability of a Type II error is denoted by β. We aim to minimise both types of errors, but reducing one often increases the other.

This test is used when we have a fixed number of trials (n), each with two possible outcomes (success/failure), and a constant probability of success (p) under the null hypothesis. We test a hypothesis about the population proportion (p). The test statistic is the number of successes (X) observed in the sample. We calculate the probability of observing a result as extreme as, or more extreme than, the sample result, assuming H₀ is true, using the binomial probability distribution. This probability is the p-value. If the p-value is less than the significance level, we reject H₀.

This test is used to determine if a sample mean (x̄) is significantly different from a hypothesised population mean (μ₀) when the population standard deviation (σ) is known, or when the sample size is large enough (n > 30) for the Central Limit Theorem to apply, allowing the sample mean to be approximated by a Normal distribution. The test statistic (z) measures how many standard errors the sample mean is away from the hypothesised population mean. We compare this z-value to critical values from the standard Normal distribution or use it to find a p-value.

This test is used to determine if there is a statistically significant linear relationship between two variables in a population, based on a sample's PMCC (r). The null hypothesis (H₀) typically states that there is no linear correlation in the population (ρ = 0), where ρ is the population PMCC. The alternative hypothesis (H₁) can be that there is a positive correlation (ρ > 0), a negative correlation (ρ < 0), or simply a non-zero correlation (ρ ≠ 0). We compare the calculated sample PMCC (r) to critical values found in statistical tables, which depend on the sample size (n) and the significance level (α). If the absolute value of r exceeds the critical value, we reject H₀.