Learn hypothesis testing with Python: Hypothesis tests

Hypothesis testing is a fundamental statistical tool that allows researchers to make inferences about population parameters based on sample data. It provides a structured methodology for determining whether there is enough evidence to support a specific hypothesis.

Definition and Purpose

At its core, hypothesis testing is a method used to assess the validity of a claim or hypothesis about a population parameter. The primary objective is to determine whether the observed data provides sufficient evidence to reject a null hypothesis (H₀) in favour of an alternative hypothesis (H₁). The null hypothesis (H0) represents the default or status quo, while the alternative hypothesis (Ha) signifies the presence of an effect or difference.

Key Components of Hypothesis Testing

1. Null and Alternative Hypotheses:

- The null hypothesis (H₀) is a statement asserting that there is no effect, difference, or relationship between variables. It serves as the benchmark for testing.

- The alternative hypothesis (H₁) represents the claim that there is an effect, difference, or relationship. It challenges the null hypothesis.

2. Test Statistic:

- A test statistic is a standardized value computed from sample data during a hypothesis test. It measures the degree of agreement between the sample data and the null hypothesis. Common test statistics include z-scores, t-scores, chi-square values, and f-values.

3. Significance Level (α):

- The significance level, denoted as α (alpha), is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold for determining statistical significance. Common significance levels are 0.05, 0.01, and 0.10.

4. P-Value:

- The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming that the null hypothesis is true. A small p-value (typically less than α) indicates strong evidence against the null hypothesis.

5. Decision Rule:

- The decision rule involves comparing the p-value to the significance level (α). If the p-value is less than α, the null hypothesis is rejected in favour of the alternative hypothesis. Otherwise, the null hypothesis is not rejected.

Types of Hypothesis Tests

1. One-Sample Tests:

- These tests involve comparing a sample statistic (e.g., mean, proportion) to a known population parameter. Examples include the one-sample t-test and the one-sample z-test.

2. Independent Two-Sample Tests:

- These tests compare the sample statistics of two independent groups to determine if there is a significant difference between them. Examples include the independent two-sample t-test and the two-sample z-test.

3. Paired Sample Tests:

- Paired sample tests, or dependent sample tests, compare the means of two related groups. They are often used in before-and-after studies. An example is the paired t-test.

4. Chi-Square Tests:

- Chi-square tests are used to examine the association between categorical variables. Examples include the chi-square test for independence and the chi-square goodness-of-fit test.

5. ANOVA (Analysis of Variance):

- ANOVA tests compare the means of three or more groups to determine if at least one group mean is significantly different from the others. Variants of ANOVA include one-way ANOVA and two-way ANOVA.

Practical Applications

Hypothesis testing finds applications across various fields, including healthcare, economics, psychology, and engineering. Here are a few examples:

1. Healthcare:

- Hypothesis tests are used to evaluate the effectiveness of new treatments or medications. For instance, a clinical trial may compare the recovery rates of patients receiving a new drug versus a placebo.

2. Economics:

- In economics, hypothesis tests assess the impact of policy changes on economic indicators. For example, a study might examine whether a tax reform leads to a significant increase in employment rates.

3. Psychology:

- In psychology, researchers use hypothesis tests to investigate the effects of interventions on behavioural outcomes. An experiment might test whether a new teaching method improves students’ test scores.

4. Engineering:

- Engineers use hypothesis tests to ensure the quality and reliability of manufacturing processes. For instance, a quality control test might determine if the defect rate in a production line exceeds a specified threshold.

Conclusion

Hypothesis testing is a cornerstone of statistical analysis, providing a rigorous framework for making data-driven decisions. By comparing observed data to theoretical expectations, researchers can determine whether there is sufficient evidence to support or reject a hypothesis. The key components, types, and applications of hypothesis tests underscore their versatility and importance in various domains. Ultimately, hypothesis testing empowers researchers and practitioners to draw meaningful conclusions, advance knowledge, and drive innovation based on empirical evidence.

See video on hypothesis tests:- https://youtu.be/EwPkxZ5oBY4

Hypothesis test practice questions

1. Check bottles ae 200 ml. H0: mean = 200 ml. Ha: mean <> 200 ml. see videos on bottles:- https://youtu.be/FIdZpwD3IRI

2. A car manufacturer wonders if an additive will increase fuel efficiency by at least 3 mpg. H0: sample mean >= 3: Ha: sample mean < 3. Perform one tailed t-test. See video on additives:- https://youtu.be/Nra7JewTqfk

3. A battery has a life span of 300 hours. Take a random sample of 30 batteries to measure their lifespans. Perform a two tailed t-test. H0: sample mean = 300 hours; Ha: sample mean <> 300 hours. See video on battery life span:- https://youtu.be/qXPeHYsUBnA

4. A software feature will reduce average response time. Current average response time is 4,5 seconds. Take a sample of 25 responses and see if the response time has reduced. Perform a one tailed left t-test. H0: sample_mean >= 4.5 seconds; Ha: sample_mean < 4.5 seconds. See video on software features:- https://youtu.be/I-AJwWYSXMs

5. A drug has a minimum of 5 mg per dose. Ensure the concentration does not fall below this level. Perform a one tailed left test to check for a significant decrease in the sample mean. H0: sample mean >= 5mg; Ha: sample mean < 5 mg. See video on drug trials:- https://youtu.be/3vLTiYZMYCs

6. The average age of male mba students is 28. Collect ages of male students across 40 mba programmes. One tailed right test. H0: sample mean <= 28; Ha: sample mean > 28. See video on male mba students:- https://youtu.be/Sbeyw2qMU4U

7. 60% of people prefer tea over coffee. A survey of 500 people show that 280 people prefer tea over coffee. Two tailed z test. H0: = 60 ; Ha: p<> 60. See video on tea/coffee survey:- https://youtu.be/QhOgz6IYKjc

8. 50 percent of people wore masks in public. A survey of 600 people shows that 330 wore masks in public. One tailed right test. H0: p <= 50%; Ha: p > 50%. See video on masks:- https://youtu.be/65qKVRA6ch4

9. A uni is thinking of having a new lunch facility. If 70% or greater students want the lunch facility then the uni will adopt the new lunch facility. One tailed left z test. H0: p >= 70%; Ha: p < 70%. See video on lunch:- https://youtu.be/rLSDM9c-nOA