What is the p-value?

• The p-value is the probability of obtaining a result equal to or more extreme than what was actually observed, assuming the null hypothesis is true.

• The p-value helps us determine if results are statistically significant.

• Common threshold: p < 0.05.

• Limitations: A small p-value does not guarantee a large effect size or practical significance.

The p-value is formally defined as:

\[ p = P(T \geq t \mid H_0) \]

where \(T\) is the test statistic and \(H_0\) is the null hypothesis.

1. Define null and alternative hypotheses.
2. Choose a significance level (\(\alpha\)).
3. Calculate the test statistic.
4. Find the p-value.
5. Compare p-value with \(\alpha\).
6. Draw conclusions.

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p_values <- replicate(1000, t.test(rnorm(30))$p.value)

ggplot(data.frame(p_values = p_values), aes(x = p_values)) + # Fixed data argument geom_histogram(binwidth = 0.05, fill = “blue”, color = “white”) + labs(title = “Distribution of p-values under the Null Hypothesis”, x = “p-value”, y = “Frequency”)

##Significance Level and p-value (LaTeX)

If \[ 𝑝 ≤ 𝛼 \] we reject the null hypothesis

Common choices for \[𝛼\] include 0.05, 0.01, and 0.10.

##Conclusion

-The p-value is a critical concept in hypothesis testing.

-It allows us to make decisions based on statistical evidence.

-Always interpret p-values in context.