Test Statistic Distribution Under Null Hypothesis

Under the null hypothesis, the test statistic is expected to follow a specific distribution that reflects the sampling behavior of the data if the null is true. This distribution is often a normal distribution, but it can vary depending on the test used, making it the primary reference for determining significance.

When testing a population mean with a large sample size, the Central Limit Theorem states that the sampling distribution of the sample mean approaches a normal distribution, regardless of the population's original distribution. Therefore, under the null hypothesis, the test statistic will also follow a normal distribution, allowing for accurate hypothesis testing.

In a two-tailed test at a significance level of α = 0.05, the critical values correspond to the points in the standard normal distribution that capture the central 95% of the data. This results in critical values of approximately ±1.96, marking the boundaries beyond which we would reject the null hypothesis.

The t-distribution is specifically designed for situations where the population standard deviation is unknown and the sample size is small, typically less than 30. It accounts for the increased variability in smaller samples, providing a more accurate estimation of confidence intervals and hypothesis testing compared to the normal distribution in these scenarios.

When the test statistic falls in the critical region, it indicates that the observed data is unlikely under the null hypothesis. This leads to the conclusion that the null hypothesis is not supported by the evidence, prompting researchers to reject it in favor of the alternative hypothesis.

In a chi-square test, the degrees of freedom are determined by the number of categories or groups being analyzed. Specifically, for a goodness-of-fit test, the degrees of freedom are calculated as the number of categories minus one (k-1). This accounts for the constraints imposed by the total sample size.

A p-value quantifies the evidence against the null hypothesis by indicating the likelihood of obtaining a test statistic at least as extreme as the observed value, assuming the null hypothesis holds true. A low p-value suggests that such an extreme outcome would be unlikely under the null hypothesis, prompting further investigation.

The F-distribution is used to compare the variances of two populations. It arises when the ratio of two independent chi-square variables is taken, making it suitable for hypothesis testing regarding variance equality. This is essential in analysis of variance (ANOVA) and other statistical tests that assess variance differences.

In a paired t-test, the null hypothesis posits that there is no significant difference between the means of two related groups. This implies that the average difference between paired observations is expected to be zero, indicating that any observed differences are due to random variation rather than a true effect.

In a one-tailed test with a significance level of α = 0.05, the critical value corresponds to the point in the standard normal distribution where 5% of the data lies in the tail. This value is approximately 1.645, indicating that any test statistic exceeding this value would lead to rejecting the null hypothesis.

In a t-test for two independent samples, degrees of freedom are calculated by summing the sizes of both samples (n₁ and n₂) and subtracting 2. This accounts for the estimation of two population parameters, allowing for a more accurate assessment of the variability between the groups being compared.

For a one-sample proportion test, the normal approximation is valid when both the expected number of successes (np₀) and failures (n(1-p₀)) are sufficiently large. Specifically, a minimum of 5 ensures that the distribution of the test statistic is approximately normal, allowing for reliable hypothesis testing.