Inferential Statistics Revision Notes for Leaving Cert Mathematics

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

Formulate Hypotheses:

Define $H_0$ and $H_1$

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

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

Calculate Test Statistic:

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

Find P-Value:

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

Make a Decision:

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

Worked Examples

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Example 1: Testing a Population Mean

Problem: A factory claims the mean weight of its products is 50 kg. A sample of 30 products has a mean weight of 49.5 kg and a standard deviation of 1.2 kg.

Test the claim at α = 0.05

Solution:

Step 1: Set Hypotheses:

$H_0: \mu = 50$
$H_1: \mu \neq 50$

Step 2: Calculate Test Statistic:

z = \frac{\bar{x} - \mu}{s / \sqrt{n}} = \frac{49.5 - 50}{1.2 / \sqrt{30}} \approx -2.28

Step 3: Find P-Value:

Using a z-table or calculator:

p = 2 \times P(Z \leq -2.28) \approx 0.0226

Step 4: Compare with $\alpha$ :

p = 0.0226 < 0.05

Step 5: Conclusion:

Reject $H_0$ .

There is strong evidence the mean weight is not 50 kg

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Example 2: Interpreting a P-Value

Problem: A p-value of 0.08 is obtained in a hypothesis test with α = 0.05.

What is the conclusion?