## 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.