Hypothesis testing is a fundamental tool in statistical analysis that allows us to make decisions about a population based on sample data. It involves formulating a hypothesis about a population parameter, collecting data, and using statistical techniques to determine the probability of obtaining the observed results if the hypothesis were true.
There are many different types of hypothesis tests, each with its own specific characteristics and uses. This post will explore the most common types of hypothesis tests and how they are used in statistical analysis.
1. Tests for Comparing Means:
One-sample z-test: This test is used to compare the mean of a sample to a known population mean. It is used when the population variance is known, or the sample size is large (n > 30).
Two-sample z-test: This test is used to compare the means of two independent samples. It is used when the population variances are known, or the sample sizes are large (n > 30).
One-sample t-test: This test is used to compare the mean of a sample to a known population mean. It is used when the population variance is unknown, and the sample size is small (n < 30).
Two-sample t-test: This test is used to compare the means of two independent samples. It is used when the population variances are unknown, and the sample sizes are small (n < 30).
- Paired t-test: This test is used to compare the means of two related samples, such as the before and after measurements of the same group of subjects. It is used when the population variances are unknown, and the sample size is small (n < 30).
2. Tests for Comparing Proportions
One-sample proportion test: This test is used to compare the proportion of a sample to a known population proportion. The normal approximation is used when both np≥10 and n(1−p)≥10 (data should have at least 10 "successes" and at least 10 "failures" ) (in some books, it is 5)
Two-sample proportion test: This test is used to compare the proportions of two independent samples. The normal approximation is used when both np≥10 and n(1−p)≥10 (data should have at least 10 "successes" and at least 10 "failures" ) (in some books, it is 5)
3. Tests for Comparing Variance
- Chi-square test for variance: This test is used to compare the variance of a sample to a known population variance.
F-test for variance: This test is used to compare the variances of two independent samples.
4. Other Common Tests
Goodness of fit test: This test is used to determine whether a sample fits a specific distribution. It is used to compare the observed frequencies of a categorical variable to the expected frequencies under a particular distribution.
Testing for independence of two attributes (Contingency Tables): This test is used to determine whether there is a relationship between two categorical variables. It is often used in the form of a chi-square test, which compares the observed frequencies in a contingency table to the expected frequencies under the assumption of independence.
ANOVA (Analysis of Variance): This test is used to compare the means of three or more independent samples. It is used to determine whether there is a significant difference between the means of the groups.
Summary
There are many different types of hypothesis tests, including tests for comparing means (such as the one-sample and two-sample t-test and z-test), proportions (such as the one-sample and two-sample proportion test), and variance (such as the chi-square and F-test). In addition, there are tests for specific data types, such as the goodness of fit test for categorical data and the ANOVA for comparing the means of multiple groups. These tests are used to make decisions about relationships between variables and to compare groups based on sample data.
This list of hypothesis tests is not exhaustive and only includes the most common tests used in statistical analysis. Many more tests are used in specific circumstances or for specific types of data. Some examples of additional tests include the Mann-Whitney U test, Wilcoxon signed-rank test, the Kruskal-Wallis test, and Spearman's rank correlation test.
In addition, there are also advanced techniques, such as bootstrapping, which can be used to make inferences about a population based on sample data. It is important to choose the appropriate test based on the data's characteristics and the study's requirements to make accurate and reliable conclusions.
