The Fundamentals of Hypothesis Testing

Let’s explore some basics of Hypothesis Testing for the beginners, with some real life business case.

Introduction to Hypothesis Testing

Well, Hypothesis testing is a statistical method used to evaluate the credibility of a hypothesis by examining the evidence that supports or refutes it. It is a critical tool used by companies and researchers around the world to determine the likelihood that their hypothesis is true or false.

The use of hypothesis testing allows us to make inferences and draw conclusions about a population based on a sample of data. This allows us to make more informed decisions and to identify trends or patterns that may not be apparent with just a small amount of data.

Hypothesis testing is a valuable tool for answering many business questions and testing ideas. It is commonly used in fields such as marketing, new feature testing by many MNC’s, and medicine to evaluate the effectiveness of treatments and interventions, as well as to identify potential trends or patterns in data.

The Basics of Hypothesis Testing

In hypothesis testing, the null hypothesis is a statement that represents the default position that there is no significant relationship between the variables being studied. The alternative hypothesis is the opposite of the null hypothesis and states that there is a significant relationship between the variables.

For example, we may have a hypothesis that a certain medication is effective in reducing symptoms of a particular illness. The null hypothesis(Ho), in this case, would be that the medication has no effect on symptoms, while the alternative hypothesis(H1) would be that the medication does have an effect on symptoms.

There are two types of errors that can be made in hypothesis testing: a Type I error and a Type II error. A Type I error occurs when the null hypothesis is rejected, even though it is true. A Type II error occurs when the null hypothesis is accepted, even though it is false.

It is important to carefully consider the potential for these types of errors when conducting a hypothesis test, as they can impact the validity and reliability of the results.

Some Important Basic Concepts You Should Know

• Null hypothesis:
  A statement that represents the default position that there is no significant relationship between the variables being studied.
  For example, Null hypothesis (H0): The marketing campaign will have no effect on sales.
  
• Alternative hypothesis:
  The opposite of the null hypothesis, states that there is a significant relationship between the variables.
  For example, Alternative hypothesis (H1): The marketing campaign will increase sales.
  
• Significance level:
  In hypothesis testing, the significance level (also called alpha or α) is a threshold that is used to determine whether the results of a statistical test are statistically significant or not. It is the probability of rejecting the null hypothesis (H0) when it is actually true.
  For example, if the significance level is set at 0.05, it means that there is a 5% chance of rejecting the null hypothesis when it is actually true.
  
• Test statistic:
  A numerical value calculated from the sample data is used to evaluate the null hypothesis.
  For Example: t-statistic is: t = (x̄ – μ) / (s / √n)
  
• P-value:
  The p-value is a way to measure the strength of the evidence against the null hypothesis. The lower the p-value, the stronger the evidence against the null hypothesis and the more likely it is that the null hypothesis is false.
  
  Example: A researcher is studying the effectiveness of a new drug for reducing blood pressure. The null hypothesis is that the drug has no effect on blood pressure, while the alternative hypothesis is that the drug reduces blood pressure. The researcher conducts a clinical trial with a sample of 50 patients, with 25 patients receiving the drug and 25 patients receiving a placebo. After four weeks of treatment, the researcher measures the blood pressure of all the patients and calculates the mean blood pressure for each group.
  
  The researcher calculates the p-value for the hypothesis test and finds that it is 0.03. The researcher has set the significance level (alpha) at 0.05.
  
  In this case, the p-value of 0.03 is less than the significance level of 0.05, which means that the results of the study are statistically significant. The researcher can conclude that the drug has a statistically significant effect on blood pressure and reject the null hypothesis. The researcher can also say that there is a 97% probability that the observed effect is real and not due to chance alone.
  
• Type I error:
  A mistake is made by rejecting the null hypothesis when it is true.
  
• Type II error:
  A mistake is made by accepting the null hypothesis when it is false.
  
• Power:
  The probability of correctly rejecting the null hypothesis when it is false.
  
• Confidence Interval:
  An interval estimate of a population parameter, such as the mean, is calculated from sample data.
  
• Sample:
  A subset of the population being studied is used to make inferences about the population.
  
• Population:
  The entire group of individuals or objects being studied.
  
• Parameters:
  Numerical characteristics of a population, such as the mean or standard deviation
  .
• Statistics:
  Numerical characteristics of a sample, such as a sample mean or sample standard deviation.
  
• Inference:
  The inference is the process of drawing conclusions or making predictions based on evidence or observations. It involves using information from a sample of data to make inferences about a larger population.
  In very simple terms, inference is a way of using information from a smaller group to make generalizations about a larger group. It is an important tool in statistical analysis because it allows us to make educated guesses about populations based on limited information.
  

Some Commonly Used Hypothesis Tests

Here are several commonly used hypothesis tests that are used in different situations and with different types of data. Some of the most commonly used hypothesis tests include:

• Z-test: This is a parametric test used to compare the means of two groups. It is often used when the sample sizes are large and the population standard deviations are known.
• T-test: This is a parametric test used to compare the means of two groups. It is often used when the sample sizes are small or the population standard deviations are unknown.
• Chi-squared test: This is a nonparametric test used to compare the distribution of categorical data between two groups. It is often used when the data is not normally distributed or the sample sizes are small.

Real-Life Case Study

Business Case
A company selling organic soap wants to determine whether its new marketing campaign has led to an increase in sales. The company has been running the campaign for the past three months and has data on soap sales during that time.

Let’s start with defining our null and alternative hypothesis

Null Hypothesis: The null hypothesis is that there is no difference in soap sales before and after the marketing campaign.

Alternative Hypothesis: The alternative hypothesis is that there is a difference in soap sales before and after the marketing campaign.

Gather data: Collect data on soap sales during the three months before the marketing campaign and during the three months of the marketing campaign.

Choose a statistical test: In this case, a paired t-test is appropriate because the samples are related (i.e., they are the same product in different time periods).

Calculate the test statistic: Find the difference in soap sales between the two periods for each month and calculate the mean of these differences. Divide this mean by the standard error of the differences to get the test statistic.

Determine the p-value: Compare the test statistic to a t-distribution with degrees of freedom equal to the number of months in the study (3). The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated, given that the null hypothesis is true.

Interpret the results: If the p-value is less than the predetermined significance level (usually 0.05), then the company can reject the null hypothesis and conclude that the marketing campaign has led to a significant increase in soap sales. If the p-value is greater than the significance level, then the company cannot reject the null hypothesis and cannot conclude that the marketing campaign has had an effect on soap sales.

Some Advanced Topics in Hypothesis Testing

There are several advanced topics in hypothesis testing that are used in more specialized situations. Some of these advanced topics include:

Power Analysis:
This is a statistical technique used to determine the sample size needed to detect a difference between the null and alternative hypotheses, given a specified significance level and power.

Multiple hypothesis testing:
This is a statistical approach used to control the error rate when testing multiple hypotheses simultaneously.

Bayesian hypothesis testing:
This is a statistical approach that uses Bayesian probability to evaluate the credibility of a hypothesis, taking into account both the evidence in support of the hypothesis and the prior probability of the hypothesis being true.

I will write on the above topics in more detail in my upcoming blogs

Conclusion

In conclusion, hypothesis testing is a statistical method used to evaluate the credibility of a hypothesis by examining the evidence that supports or refutes it. It is a critical tool used by researchers and scientists to determine the likelihood that their hypothesis is true or false and to make more informed decisions based on the results of the tests.

The steps involved in conducting a hypothesis test include defining the research question or hypothesis, choosing a significance level, determining the sample size, collecting and analyzing data, calculating the test statistic, and making a decision about the null hypothesis.

There are several commonly used hypothesis tests, such as the z-test, t-test, and chi-squared test, as well as several advanced topics, such as power analysis, multiple hypothesis testing, and Bayesian hypothesis testing.

Overall, hypothesis testing is an important statistical tool that is widely used in many fields to answer many business questions and test theories.

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