This slideshow will cover the p-value and its significance in statistics, mainly hypothesis testing.
2025-03-16
P-Value in Statistics
What is a P-Value?
A p-value measures the probability of obtaining an extreme result under the null hypothesis.
Lower p-values suggest stronger evidence against \(H_0\).
Key rule: If \(p < \alpha\), typically 0.05, we reject \(H_0\).
What is Hypothesis Testing?
Hypothesis testing determines if there is enough evidence to reject the null hypothesis.
It involves:
- Defining hypotheses:
- \(H_0\): Null hypothesis (no effect).
- \(H_1\): Alternative hypothesis (some effect).
- Collecting sample data.
- Computing a test statistic.
- Comparing the p-value to a significance level (\(\alpha\)).
- Defining hypotheses:
The p-value is crucial in hypothesis testing.
Mathematical Derivation of the P-Value
The test statistic for a one-sample t-test is:
\[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]
where: \(\bar{x}\) is the sample mean, \(\mu_0\) is the hypothesized population mean, \(s\) is the sample standard deviation, \(n\) is the sample size.
3D Visualization of P-Values
This 3D plot (plotly) shows how sample size affects p-values.
When a sample size is small, p-values tend to be large, meaning we fail to reject more often
Simple Line Plot of P-Values
A simple line plot showing how p-values decrease as sample size increases.
Simple Histogram of Simulated P-Values
A histogram showing the distribution of p-values from 100 t-tests.
library(ggplot2)
# Generate random p-values from simulated t-tests
set.seed(123)
p_values <- replicate(100, t.test(rnorm(30))$p.value)
data <- data.frame(p_values)
ggplot(data, aes(x = p_values)) +
geom_histogram(binwidth = 0.05, fill = "blue", color = "black") +
labs(title = "Histogram of Simulated P-Values",
x = "P-Value",
y = "Frequency")
