## 3.1 Effect size for a one-sample \(t\)-test

Recall the cholesterol example where we tested the hypotheses

\[H_0:\mu = 5\;\;\text{versus}\;\;H_1:\mu \neq 5,\]

where:

- \(\mu_0\) denotes the mean under the null hypothesis,

where the associated \(t\)-test results were:

```
One Sample t-test
data: heartattack$cholesterol
t = 2.2063, df = 71, p-value = 0.0306
alternative hypothesis: true mean is not equal to 5
95 percent confidence interval:
5.012405 5.245325
sample estimates:
mean of x
5.128865
```

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

```
Cohen's d (single sample)
d estimate: 0.2600188 (small)
Reference mu: 5
95 percent confidence interval:
lower upper
-0.2119399 0.7319776
```

As we can see, the effect size was 0.26 (this can be thought of as 0.26 standard deviations) and is considered small.