## 3.2 Effect size for an independent samlpes $$t$$-test

Recall the cholesterol example where we tested the hypotheses

$H_0:\mu_1 = \mu_2\;\;\text{versus}\;\;H_1:\mu_1 \neq \mu_2,$ where:

• $$\mu_1$$ denotes the true average cholesterol level of patients in the high risk group
• $$\mu_2$$ denotes the true average cholesterol level of patients in the low risk group,

where the associated $$t$$-test results were:


Two Sample t-test

data:  heartattack$cholesterol by heartattack$risk
t = 7.5483, df = 70, p-value = 1.238e-10
alternative hypothesis: true difference in means between group high and group low is not equal to 0
95 percent confidence interval:
0.4851270 0.8335547
sample estimates:
mean in group high  mean in group low
5.458536           4.799195 

The results of the associated effect size calculation are as follows:


Cohen's d

d estimate: 1.779145 (large)
95 percent confidence interval:
lower    upper
1.223784 2.334506 

As we can see, the effect size was 1.78 (this can be thought of as 1.78 standard deviations) and is considered a large effect.