# How To Perform Hypothesis Testing

Performing hypothesis testing in the context of the Singapore Junior College (JC) math syllabus involves applying statistical methods to test hypotheses or claims about populations, sample data, or relationships between variables. It’s an essential concept in statistics and probability and has practical applications in various fields. Here’s a general guide on how to perform hypothesis testing and why it’s important in the JC math syllabus.

## Steps for Performing Hypothesis Testing:

State the Hypotheses (Null and Alternative): Begin by clearly defining the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis typically represents the status quo or no effect. (It states that a population parameter has a particular value), while the alternative hypothesis represents the claim you want to test (a range of values that you want to test).

1. Select the Significance Level (α): Choose the significance level, denoted as α, which represents the probability of making a Type I error (rejecting a true null hypothesis)/ or put simply the probability of rejecting a null hypothesis when it is true. Common values for α include 0.05 or 0.01, a value that is small but not 0! It can vary depending on the context.

Collect Data and Calculate the Test Statistic: Collect relevant data from a sample and calculate the test statistic based on the chosen hypothesis test (e.g., z-test, t-test, chi-squared test, etc.). The test statistic quantifies how far the sample data deviates from what is expected under the null hypothesis. This step can be done on the GC calculator

Determine the Critical Region: Determine the critical region (rejection region) by finding the critical values from the appropriate statistical distribution. The critical region represents extreme values of the test statistic that would lead to the rejection of the null hypothesis if observed.

Compare Test Statistic and Critical Values: Compare the calculated test statistic with the critical values. If the test statistic falls within the critical region, reject the null hypothesis. If it falls outside the critical region, fail to reject the null hypothesis.

Draw and state the Conclusion: Based on the comparison, make a statistical decision about whether to reject or fail to reject the null hypothesis. This decision should be communicated along with the reasoning and any practical implications. Finally, state the conclusion of the hypothesis test in the context of the problem, using appropriate language to convey the statistical findings.

## Why Hypothesis Testing is Important in JC Math:

Skills acquired from hypothesis vary but are added to the syllabus as it is used in many forms of decision making such as:

• Statistical Reasoning: Hypothesis testing develops critical statistical reasoning skills, which are crucial for students studying JC math. It teaches students to analyze data and make informed decisions based on evidence.

Similarly, performing hypothesis tests requires students to formulate clear hypotheses, choose appropriate test statistics, and make decisions based on statistical evidence. This can be used in fields such as scientific Inquiry, which values how scientific research and experiments are conducted and evaluated.

Data-Driven Decision Making: In today’s data-driven world, the ability to assess hypotheses and make decisions based on data is essential. Understanding hypothesis testing equips students with the skills needed to evaluate claims and make informed choices.

Preparation for Higher Education and Careers: Proficiency in hypothesis testing is valuable for students pursuing further education in fields such as mathematics, science, social sciences, economics, and engineering. It’s also relevant in careers involving data analysis and research.

Overall, hypothesis testing is a foundational concept in statistics that fosters critical thinking and data analysis skills, preparing students for both academic and real-world challenges in various domains.

## A Real life case study of Hypothesis study! (Covid-19)

During the COVID-19 pandemic, researchers around the world raced to develop effective vaccines to combat the virus caused by SARS-CoV-2. One of the key phases in vaccine development involves conducting clinical trials to determine whether a new vaccine is safe and effective.

Hypothesis Testing in Vaccine Development (Follow along the steps as taught above):

Formulating Hypotheses:

Null Hypothesis (H0): The COVID-19 vaccine has no significant effect in preventing the disease.

Alternative Hypothesis (Ha): The COVID-19 vaccine is effective in preventing the disease.

Selecting Significance Level (α):

In clinical trials, a common significance level chosen is α = 0.05, which means researchers are willing to accept a 5% chance of making a Type I error (rejecting a true null hypothesis).

Collecting Data:

Researchers conducted large-scale clinical trials involving thousands of participants. Participants were divided into two groups: one receiving the vaccine, and the other receiving a placebo.

Calculating the Test Statistic:

In this context, the test statistic involved comparing the number of COVID-19 cases among the vaccinated group to the number of cases in the placebo group.

Determining the Critical Region:

Researchers used statistical methods to establish a critical region based on the chosen significance level (α). For example, if the p-value (probability of observing results as extreme as those obtained, assuming the null hypothesis is true) was less than α, the critical region would be reached.

Comparing Test Statistic and Critical Values:

Researchers compared the observed results to the critical values. If the number of COVID-19 cases in the vaccinated group was significantly lower than in the placebo group, the null hypothesis (H0) would be rejected.

Drawing a Conclusion:

Based on the comparison, researchers made a statistical decision. If the data provided strong evidence that the vaccine was effective in preventing COVID-19, they would reject the null hypothesis.

Stating the Conclusion:

Researchers communicated the findings, concluding that the COVID-19 vaccine was effective in preventing the disease based on the results of the clinical trials.

## Understanding the significance of the results

The breakthrough in this context occurred when multiple COVID-19 vaccines, including those developed by Pfizer-BioNTech, Moderna, and others, demonstrated high efficacy in preventing COVID-19 during clinical trials. This outcome was based on rigorous hypothesis testing, where the null hypothesis was rejected in favor of the alternative hypothesis, providing statistical evidence that the vaccines were effective.

These breakthroughs marked a significant milestone in the fight against the COVID-19 pandemic, as the vaccines became a critical tool in preventing the spread of the virus and reducing the severity of illness. Hypothesis testing in vaccine development ensures that decisions about the safety and efficacy of vaccines are based on rigorous scientific evidence, giving the public confidence in their effectiveness and safety.