P-Values

Overview

A p-value is a statistical measure used in hypothesis testing to determine the strength of evidence against the null hypothesis. It represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true.

Key Concepts

Null Hypothesis ($H_0$):

A statement that there is no effect or no difference. For example, $H_0$: The mean of the population is equal to 50.

Alternative Hypothesis ($H_1$):

A statement that contradicts the null hypothesis. For example, $H_1$: The mean of the population is not equal to 50.

P-Value Interpretation:

• Small p-value ($< \alpha$): Strong evidence against $H_0$, leading to its rejection in favour of $H_1$
• Large p-value ($\geq \alpha$): Weak evidence against $H_0$, so it is not rejected.
• Common significance levels ($\alpha$) are 0.05 or 0.01

Calculation:

The p-value depends on the statistical test used (e.g., z-test, t-test) and is typically computed using software or statistical tables.

Hypothesis Testing Procedure

1. Formulate Hypotheses:

• Define $H_0$ and $H_1$

2. Select Test and Significance Level ($\alpha$):

• Choose an appropriate test (e.g., z-test, t-test) and significance level ($\alpha$)

3. Calculate Test Statistic:

• Compute the value of the test statistic based on sample data.

4. Find P-Value:

• Use statistical software, tables, or calculators to determine the p-value.

5. Make a Decision:

• Compare the p-value with $\alpha$ and decide to reject or fail to reject $H_0$

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Worked Examples

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Summary

• P-value measures the probability of observing data as extreme as the sample, assuming $H_0$ is true.
• Interpretation:
  • $p < \alpha$: Reject $H_0$
  • $p \geq \alpha$: Fail to reject $H_0$
• Steps:
  1. Formulate $H_0$ and $H_1$
  2. Choose a test and $\alpha$
  3. Compute the test statistic and p-value.
  4. Make a decision based on $p$ and $\alpha$
• P-values are central to hypothesis testing in inferential statistics.