Hypothesis testing is a statistical method used to make decisions or inferences about a population based on a sample of data.

The goal of hypothesis testing is to determine whether there is enough statistical evidence in the sample data to reject the null hypothesis in favor of the alternative hypothesis.

A good example of this would be the following scenario. Does a new drug improve recovery rates compared to the standard?

• Null Hypothesis (H₀): A statement of no effect, no difference, or status quo.

  \(H_0: \mu =\mu_0\)

• Alternative Hypothesis (H₁ or Ha): A statement that contradicts the null hypothesis, representing a new effect or difference.

  \(H_a: \mu \neq \mu_0\)

In our example the Null Hypothesis and the Alternative Hypothesis can be the following:

• Null Hypothesis: The recovery rate of the new drug is the same as that of the standard drug.

• Alternative Hypothesis: The new drug improves recovery rates compared to the standard drug.

There are several number of tests that can be devised for Hypothesis testing.

The two most common hypothesis testing methods are the one-sided and two-sided test

The Chi-square tests compare the size of the discrepancies between the expected results and the actual results, given the size of the sample and the number of variables in the relationship.

For our example we will perform a chi-square test.

\(\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\)

• The observed frequency is:

  \(O_i\)

• The expected frequency is:

  \(E_i\)

• Expected frequency is calculated as:

  \(E_i = \frac{(\text{Row Total} \times \text{Column Total})}{\text{Grand Total}}\)

# Create a contingency table
recovery_table <- table(drug_recovery_data$Drug_Type, drug_recovery_data$Recover_Rate)
colnames(recovery_table) <- c("Not Recovered", "Recovered")
#View the table
print(recovery_table)

##           
##            Not Recovered Recovered
##   New                  3        47
##   Standard            14        36

#Perform a Chi-square test
Chi_test <- chisq.test(recovery_table)
#Output the test results
print(Chi_test)

## 
##  Pearson's Chi-squared test with Yates' continuity correction
## 
## data:  recovery_table
## X-squared = 7.0872, df = 1, p-value = 0.007764

• A p-value is a statistical measurement used to validate a hypothesis against observed data.

• The lower the p-value the greater the statistical significance of the observed difference.

If the p-value < 0.05 then we reject the null hypothesis and conclude that the new drug has a significantly higher recovery rate.

Since the p-value is shown as 0.007764 we can safely reject the Null Hypothesis in this scenario and proceed with the Alternative hypothesis.