In statistics, we often ask a simple question:

Is what we see in the data real, or could it just be random chance?

The p-value helps us answer this question.

We will begin with two statements:

\[ H_0: \mu = \mu_0 \]

\[ H_a: \mu \neq \mu_0 \]

• \(H_0\) says there is no effect
• \(H_a\) says there is an effect
• \(\mu\) is the true population mean

To test the hypotheses, we compute:

\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

This value tells us
how far the sample mean is from the null value, measured in standard error units.

The p-value is:

\[ \text{Probability of seeing results this extreme if } H_0 \text{ is true} \]

• Small p-value → strong evidence against \(H_0\)
• Large p-value → weak evidence against \(H_0\)

We will generate sample data from a normal distribution.

The goal is to test
whether the population mean is equal to 50.

This plot shows:

• The histogram of the data
• A smooth curve showing the overall shape

It helps us see where the data values fall.

The shaded areas show the p-value.

They represent outcomes that are
as extreme or more extreme than our data, assuming the null hypothesis is true.

R can calculate the p-value directly
using a one-sample t-test.

## 
##  One Sample t-test
## 
## data:  data
## t = 3.1361, df = 39, p-value = 0.00325
## alternative hypothesis: true mean is not equal to 50
## 95 percent confidence interval:
##  50.79029 53.66154
## sample estimates:
## mean of x 
##  52.22592

This plot shows how the p-value changes as the sample mean moves away from the null value and as the sample size increases. Smaller p-values mean stronger evidence against the null hypothesis.

• The p-value measures evidence against the null hypothesis
• It depends on both effect size and sample size
• A small p-value does not always mean a big or important effect

Context always matters.