# ML Wiki

## One-Sample $t$-test

This is a t-test for one variable

• it can be used to calculate a Confidence Interval for the true mean $\mu$
• the null value for $H_0$ might come from other research or from your knowledge

Parameters:

• $\text{df} = n - 1$, where $n$ is the sample size

## Examples

### Example 1

• Sample: $n = 60, \bar{X} = 7.177, s = 2.948$
• True mean $\mu$ is unknown

Let's run a test:

• $H_0: \mu = 0, H_A: \mu > 0$ (this is one-sided test)
• Under $H_0$, we know that $\cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \approx t_{n - 1}$
• Observed: $\bar{X} - \mu = 7.177 - 0 = 7.177$
• How plausible is the observed value under $H_0$?

The probability of observing this value is

• $P(\bar{X} - \mu \geqslant 7.177) =$
• $P\left(\cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \geqslant \cfrac{7.177}{\sqrt{s^2 / n}}\right) \approx$
• $P\left(t_{59} \geqslant \cfrac{7.177}{\sqrt{2.948^2 / 60}}\right) \approx$
• $P(t_{59} \geqslant 18.86) \approx 1 / 10^{26}$

Extremely small! So we reject $H_0$ and conclude that $\mu > 0$

### Example 2

• Sample $n = 400, \bar{X} = -14.15, s = 14.13$
• Test: $H_0: \mu = 0$ vs $H_A: \mu \neq 0$ (this is a 2-sided)

We know that

• $\cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \approx t_{n - 1} = t_{399}$

$p$-value:

• $P( | \bar{X} - \mu | \geqslant | -14.15 - 0 |) =$
• $P\left( \left| \cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}} \right| \geqslant \cfrac{14.15}{\sqrt{14.13^2 / 400}}\right) \approx$
• $P( | t_{399} | \geqslant 20.03 ) =$
• $2 \cdot P( t_{399} \leqslant -20.03) \approx$
• $1 / 3.5 \cdot 10^{64}$

Extremely small! Reject the $H_0$ and conclude that $\mu \neq 0$

### R code

Our test statistic is $T = \cfrac{\bar{X} - \mu}{\sqrt{s^2 / n}}$.

xbar = mean(ch)
s2 = var(ch)
n = length(ch)
mu = 0

t = (xbar - mu) / sqrt(s2 / n) // 18.856
pt(t, df=n-1, lower.tail=F) // 5.84E-24
// so we reject