🎓 Achieve Statistical Excellence: Mastering Hypothesis Testing from Basics to Advanced 📊
Mastering Hypothesis Testing: Your Complete Guide to Understanding Types, Techniques, and Real-World Examples.
I first learned about hypothesis testing at Innomatic Research Labs under the guidance of Nagaraju Ekkirala. Ever since, I’ve felt that I was missing something crucial.
Determined to understand it fully, I recently dove deep into the topic. Thanks to this exploration, I had an aha moment where everything finally made sense.
I wrote this article to explain hypothesis testing the way I wish it had been explained to me from the start. I hope it helps you reach the same aha moment I experienced.
Overview:
This article is structured to provide a comprehensive understanding of hypothesis testing, covering the following key aspects:
- Test Name
- Definition
- Purpose
- Types
- Hypothesis
- Statistical Formulas
- Assumptions
- Number of Samples Required
- Real-Life Example
- Visualization
Hypothesis Testing
Definition:
- Hypothesis testing is a statistical method used to test a specific statement or claim about a population parameter.
- It involves comparing observed data against a null hypothesis to determine the likelihood that the null hypothesis is true.
Approaches of Hypothesis Testing
Formulate Hypotheses:
- Null Hypothesis (H₀): A statement of no effect or no difference, which we assume to be true initially.
- Alternative Hypothesis (H₁ or Ha): A statement that contradicts the null hypothesis, representing the effect or difference we are testing for.
Select Significance Level (α):
- The probability threshold (commonly 0.05) for rejecting the null hypothesis, representing the risk of a Type I error (false positive).
Choose the Test Statistic:
- A value calculated from sample data used to assess the plausibility of the null hypothesis (e.g., t-test, z-test, chi-square test).
Determine the Decision Rule:
- Define the critical region based on the significance level and the test statistic’s distribution, which indicates when to reject H₀.
Collect Data and Compute Test Statistic:
- Gather sample data and calculate the test statistic.
Make a Decision:
- Compare the test statistic to the critical value or use the p-value approach:
- Reject H₀ if the test statistic falls in the critical region or if the p-value ≤ α.
- Fail to Reject H₀ if the test statistic does not fall in the critical region or if the p-value > α.
Draw Conclusion:
- Based on the decision, conclude whether there is sufficient evidence to support the alternative hypothesis.
Types of Hypothesis Testing
1. Z-Test
Definition:
- The Z-test is used to determine whether a sample mean differs significantly from a population mean when the population standard deviation is known (for 1-sample Z-test) or when comparing two sample means (for 2-sample Z-test).
Purpose:
- To compare means of large samples.
Types:
Hypothesis:
- Null (H₀): μ1 = μ2
- Alternative (H₁): μ1 ≠ μ2
One Sample Z-Test:
- Null Hypothesis (H₀): The sample mean equals the population mean.
- Alternative Hypothesis (H₁): The sample mean does not equal the population mean.
Two Sample Z-Test:
- Null Hypothesis (H₀): The means of two populations are equal.
- Alternative Hypothesis (H₁): The means of two populations are not equal.
Statistical Formulas:
Assumptions:
- The sample size should be n ≥ 30.
- Population variance is known.
- Data should be normally distributed.
Number of Samples Required:
- n ≥ 30
Real-life Example:
- Comparing the average height of men in two different countries to see if there is a significant difference.
Visualization:
2. T-Test
Definition:
- The T-test is used to determine whether there is a significant difference between the means of two groups (for 2-sample T-test) or between a sample mean and a population mean (for 1-sample T-test), when the population standard deviation is unknown and estimated from the sample.
Purpose
- To compare means of small samples.
Types:
Hypothesis:
- Null (H₀): μ1 = μ2
- Alternative (H₁): μ1 ≠ μ2
One Sample T-Test:
- Null Hypothesis (H₀): The sample mean equals the population mean.
- Alternative Hypothesis (H₁): The sample mean does not equal the population mean.
Two Sample T-Test:
- Null Hypothesis (H₀): The means of two populations are equal.
- Alternative Hypothesis (H₁): The means of two populations are not equal.
Statistical Formulas:
Assumptions:
- Samples should be normally distributed.
- Homogeneity of variances.
Number of Samples Required:
- n < 30
Real-life Example:
- Comparing the test scores of a small class before and after implementing a new teaching method.
Visualization:
3. Chi-Square Test
Definition:
- A Chi-square test is used to compare categorical data frequencies.
Purpose:
- To compare categorical data frequencies.
Types:
Hypothesis:
Goodness of Fit Chi-Square Test:
- Null Hypothesis (H₀): Observed frequencies fit the expected frequencies.
- Alternative Hypothesis (H₁): Observed frequencies do not fit the expected frequencies.
Independence Chi-Square Test:
- Null Hypothesis (H₀): There is no association between the categorical variables.
- Alternative Hypothesis (H₁): There is an association between the categorical variables.
- Null (H₀): o = e
- Alternative (H₁): o ≠ e
Statistical Formulas:
Assumptions:
- Expected frequency in each cell should be at least 5.
Number of Samples Required:
- n > 5
Real-life Example:
- Testing if the distribution of different blood types in a population matches expected proportions.
Visualization :
4. ANOVA Test
Definition:
- ANOVA (Analysis of Variance) is used to compare the means of three or more groups.
Purpose:
- To compare means among three or more groups.
Types:
Hypothesis:
- Null (H₀): μ1 = μ2 = μ3 … = μk
- Alternative (H₁): At least one μ is different
One-Way ANOVA:
- Null Hypothesis (H₀): All group means are equal.
- Alternative Hypothesis (H₁): At least one group mean is different.
Two-Way ANOVA:
- Null Hypothesis (H₀): There is no interaction between the two factors, and the means of all groups are equal.
- Alternative Hypothesis (H₁): There is an interaction between the factors, or the means of at least one group are different.
Statistical Formulas:
Assumptions:
- Samples should be normally distributed.
- Homogeneity of variances.
- Independence of observations.
Number of Samples Required:
- At least 30 per group per level
Real-life Example:
- Comparing the average yield of different crop varieties across multiple farms.
Visualization:
5. Shapiro-Wilk Test
Definition:
- A Shapiro-Wilk test is used to assess the normality of a distribution.
Purpose:
- To test for normality of data.
Types:
Hypothesis:
- Null (H₀): Data is normally distributed
- Alternative (H₁): Data is not normally distributed
Statistical Formulas:
Assumptions:
- Applicable to small sample sizes (n < 50), can be used up to n = 2000.
Number of Samples Required:
- n < 2000
Real-life Example:
- Checking if a set of test scores follows a normal distribution.
Visualization:
6. Kolmogorov-Smirnov (KS) Test
Definition:
- A KS test is used to compare two sample distributions.
Purpose:
- To compare two sample distributions.
Types:
Hypothesis:
- Null (H₀): No difference between the distributions
- Alternative (H₁): Distributions differ
Statistical Formulas:
Assumptions:
- Data is continuous.
Number of Samples Required:
- Flexible based on data.
Real-life Example:
- Comparing the distribution of income levels in two different cities.
Visualization:
7. Mann-Whitney U Test
Definition:
- A Mann-Whitney U test is a non-parametric method used to compare differences between two independent samples.
Purpose:
- To compare differences between two independent samples.
Types:
Hypothesis:
- Null (H₀): Distributions of both samples are equal
- Alternative (H₁): Distributions are not equal
Statistical Formulas:
Assumptions:
- Data is ordinal, interval, or ratio scale.
- Samples are independent.
Number of Samples Required:
- Flexible based on data.
Real-life Example:
- Comparing customer satisfaction ratings between two stores.
Visualization:
8. Omnibus Test
Definition:
- A statistical test used to assess overall significance across multiple groups or conditions.
Purpose:
- To test for overall significance across multiple groups or conditions.
Types:
Hypothesis:
- Null (H₀): No effect or difference among groups
- Alternative (H₁): At least one group is different
Statistical Formulas:
- Varies based on the specific test (e.g., ANOVA F-statistic, Chi-square statistic)
Assumptions:
- Varies based on the specific test used.
Number of Samples Required:
- Varies based on the specific test used.
Real-life Example:
- Assessing the effectiveness of different marketing strategies across several regions.
Visualization:
9. Correlation Coefficient Test
Definition:
- A statistical test used to measure the strength and direction of the relationship between two variables.
Purpose:
- To measure the strength and direction of the relationship between two variables.
Types:
Hypothesis:
- Null (H₀): ρ = 0 (no correlation)
- Alternative (H₁): ρ ≠ 0 (correlation exists)
Statistical Formulas:
Assumptions:
- Pearson: Linear relationship, interval or ratio scale, normal distribution.
- Spearman: Ordinal data or non-linear relationships.
Number of Samples Required:
- Flexible based on data.
Real-life Example:
- Examining the relationship between hours studied and exam scores.
Visualization:
Comprehensive Summary of Hypothesis Tests: Parameters and Features
Contact
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