Chapter 4 Dive into the t-test

After talking about p-values and hypothesis tests, you’re probably wondering, “How do we actually test these hypotheses?” Enter the t-test, a powerful tool that helps us compare means and decide whether observed differences are statistically significant.

4.1 Basics of the t-test

The t-test helps us determine whether two groups have different means. This test assumes that the data follows a normally distributed pattern when the sample size is small and that variances are equal, unless stated otherwise.

There are mainly two types of t-tests: 1. Independent samples t-test: Used when comparing the means of two separate groups, like testing a new teaching method by comparing test scores from two different classrooms. 2. Paired sample t-test: Useful when comparing measurements taken from the same group at different times, such as before and after a specific treatment in a medical study.

4.2 Step-by-Step Example Using Simulated Data

Let’s consider a scenario where we’re testing a new fertilizer on plant growth. We have a control group (old fertilizer) and a treatment group (new fertilizer). We want to know if the new fertilizer leads to better plant growth.

4.2.1 Setting Up the Problem

# Simulating plant heights for control and treatment groups
set.seed(42)
control <- rnorm(30, mean=20, sd=5)  # Control group, N=30, mean height = 20 cm
treatment <- rnorm(30, mean=23, sd=5)  # Treatment group, N=30, mean height = 23 cm

4.2.2 Performing an Independent Samples t-test

# Comparing the two groups
t_test_result <- t.test(control, treatment, alternative = "two.sided")
t_test_result

## 
##  Welch Two Sample t-test
## 
## data:  control and treatment
## t = -1.3707, df = 56.249, p-value = 0.1759
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -5.0396655  0.9446189
## sample estimates:
## mean of x mean of y 
##  20.34293  22.39046

The output of this t-test will provide us with a p-value, which tells us if the differences in plant growth are statistically significant.

4.3 Interpreting Results

If our p-value is less than 0.05, we can reject the null hypothesis and conclude that the new fertilizer makes a significant difference in plant growth. If it’s higher, we might need more data or accept that the new fertilizer doesn’t significantly outperform the old one.

Awesome! Let’s tackle A/B testing next. That section will show how A/B testing is a practical application of hypothesis testing and t-tests in real-world decision-making scenarios.