1. Review of Previous Content on Hypothesis Testing
Building on our previous detailed introduction to descriptive statistics,
▶︎ 2.1 Descriptive statistics vs Inferential statistics
▶︎ 2.2 Parameter Estimation_Inferential Statistics (1)
▶︎ 2.3 Probabilities and Significant level_Inferential Statistics (2)
We have already covered concepts such as two-tailed tests, one-tailed tests, and significance levels. Next, we will formally introduce hypothesis testing in inferential statistics, focusing on methods for inferring population parameters from samples.
The entire learning process will be divided into three parts:
▶︎ Estimating the population mean from a single sample
▶︎ Comparing the means of two independent samples
▶︎ Comparing the means of two related samples
Today, we will comprehensively introduce the two main types of hypothesis testing: parametric vs non-parametric tests, establishing an overall framework for understanding hypothesis testing. We will discuss the factors to consider in parametric analysis and how to determine the specific hypothesis test to choose.
In the next template, we will start with an example of Parametric Hypothesis Test for Sample vs Population. We will use a real-world example and Bayeslab's AI visualization tools to help you better and more deeply understand statistical analysis methods.
Throughout this process, we will use the Bayeslab online AI analysis tool. With the help of AI Agent's visualization capabilities, we can obtain Python analytical code tailored by the AI Agent with just simple prompts. This approach enables us to efficiently perform Python data visualization and routine business analysis. The outcome includes not only a set of visualized images but also Python source code, data analysis files, and even assists individuals with no statistical or Python coding background to complete professional data analysis.
OK, let's get started.
2. What is hypothesis test in statistics?
Hypothesis test definition:
Hypothesis testing is one of the fundamental tasks of inferential statistics, which involves using sample information to evaluate the likelihood of a hypothesis about population parameters (population distribution) based on certain probabilities.
This leads to a decision to either reject or retain the hypothesis.
Hypothesis testing is divided into parametric and nonparametric hypothesis testing.
2.1 What is a parametric test of hypothesis?
A parametric test of hypothesis is a type of statistical test that makes specific assumptions about the parameters and the distribution of the population from which the sample data is drawn.
These tests rely on the assumption that the data follows a particular distribution, usually a normal distribution, and involve parameters such as the mean and standard deviation.
Key features of parametric tests include:
● Assumption of Data Distribution: They assume that the data follows a known and specific distribution (e.g., normal distribution).
● Parameter Estimation: They involve estimating population parameters (e.g., mean, standard deviation) from the sample data.
● Statistical Efficiency: Under their assumptions, parametric tests tend to be more statistically powerful and efficient, meaning they can detect true effects with smaller sample sizes compared to non-parametric tests.
Common examples of parametric tests include:
● t-test: Used to compare the means of two groups (e.g., independent samples t-test, paired samples t-test).
● Z-test: Used for testing the means of large sample sizes or the proportions of large sample sizes.
● ANOVA (Analysis of Variance): Used to compare the means of three or more groups.
● F-test: Used to compare the variances of two populations.
Overall, parametric tests are preferred when the assumptions about the population distribution are met, as they provide more accurate and powerful results.
2.2 What is a non parametric hypothesis testing?
Non-parametric hypothesis testing is a type of statistical test that does not require assumptions about the specific distribution of the population from which the sample is drawn. These tests are used when the data do not meet the assumptions necessary for parametric tests, such as normality, or when dealing with ordinal data, ranks, or non-quantitative categories.
Key features of non-parametric hypothesis tests include:
● No Distribution Assumption: They do not assume a specific probability distribution for the data.
● Flexibility: They can be applied to a wide range of data types, including ordinal, nominal, and interval data without standard distribution requirements.
● Robustness: They are more robust to outliers and deviations from typical distribution assumptions.
● Rank-Based Methods: They often use data ranking instead of actual data values, which makes them applicable to data that do not meet the criteria for parametric tests.
Common examples of non-parametric tests include:
● Mann-Whitney U test: Used to compare differences between two independent groups.
● Wilcoxon Signed-Rank test: Used for comparing paired data.
● Kruskal-Wallis test: Used for comparing three or more independent groups.
● Friedman test: Used for comparing three or more related groups.
● Chi-squared test: Used for testing relationships between categorical variables.
You can find practical examples of all the relevant testing methods mentioned above in the Library Graph Portfolio category of Bayeslab. These are useful data analysis cases I have prepared previously, which can be executed with a prompt. You can also replace the data in these prompts to perform further analysis and modifications.
2.3 What is the difference between parametric and nonparametric hypothesis testing?
Let's summarize the difference we all we mentioned with the table below:
3. Which hypothesis test to use?
Similar to parameter estimation, the formulas for parametric hypothesis testing and sampling also correspond to different situations and require consideration of the distribution the data might follow under various factors. For example:
Whether the population variance (σ²) is known or unknown ?
Whether the population follows a normal distribution X~(μ,σ²) ?
Whether the sample size is large or small ?
Whether sampling method is with or without replacement ?
The key difference between parametric hypothesis testing and parameter estimation is the inclusion of a null hypothesis and an alternative hypothesis, which also requires consideration of whether to use a two-tailed or one-tailed test, as introduced in the previous template.
Except for whether sampling is with or without replacement, where we add a correction factor for the result, the main distributions involved in inferential statistics for parametric and non-parametric hypothesis testing, after considering the other factors, are the
t-distribution,
Z-distribution,
chi-square distribution,
F-distribution.
We will introduce each of these in the simplest way possible in the following sections.
This is the basic framework of hypothesis testing. In the following templates, we will outline specific steps using real data sources. Don't worry if you're not familiar with statistical analysis or Python; with Bayeslab's prompts, you can still complete the exercises. If you have time, join us in the next template. Feel free to subscribe!
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