Two-sample inference for the difference between groups(KA/SAP/U13TwoSampleInference)

Unit 13: Two-sample inference for the difference between groups

Comparing Two Proportions

What is it? Comparing two proportions involves determining whether the proportion of a certain outcome is different between two groups. For example, you might want to know if the proportion of people who prefer tea is different between two cities.

• Null hypothesis (H₀): The proportions are equal (p₁ = p₂).
• Alternative hypothesis (H_a): The proportions are not equal (p₁ ≠ p₂), or one is greater/less than the other.
• Test statistic: Uses the difference between sample proportions and the standard error.
• Conditions: Random samples, independence, and large enough sample sizes (np ≥ 10 and n(1-p) ≥ 10 for both groups).

Quiz: Comparing Two Proportions

Suppose 60 out of 100 people in Group A like chocolate, and 45 out of 100 in Group B like chocolate. Is there evidence that the proportions are different?

Yes
No

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Comparing Two Means

What is it? Comparing two means is about determining if the average value of a variable differs between two groups. For example, do students in School A score higher on a test than those in School B?

• Null hypothesis (H₀): The means are equal (μ₁ = μ₂).
• Alternative hypothesis (H_a): The means are not equal (μ₁ ≠ μ₂), or one is greater/less than the other.
• Test statistic: Uses the difference between sample means and the standard error (often a t-test).
• Conditions: Random samples, independence, and normality (or large enough samples for the Central Limit Theorem).

Quiz: Comparing Two Means

The average score in Group A is 75 (SD=10, n=30), and in Group B is 80 (SD=12, n=30). Is there likely a significant difference?

Yes
No

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Summary Table

	Comparing Proportions	Comparing Means
Parameter	p1 - p2	μ1 - μ2
Statistic	\( \hat{p}_1 - \hat{p}_2 \)	\( \bar{x}_1 - \bar{x}_2 \)
Test	z-test for proportions	t-test for means
Conditions	Random, independent, large n	Random, independent, normal/large n