ENVE-GEOE 224 - 25 - Hypothesis Testing III - Two Means & Paired Comparison - Handout

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Oct 30, 2023

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2021-03-28 1 ENVE/GEOE 224: Probability & Statistics Hypothesis Testing III – Two Means & Paired Comparison Prof. Philip J. Schmidt Department of Civil & Environmental Engineering, University of Waterloo Winter 2021 Learning Objectives Learn about hypothesis tests and which to use → Tests on the difference between two means (variances known) → Tests on the difference between two means (variances unknown and equal) → Tests on the difference between two means (variances unknown and unequal) → Tests on the mean difference in a paired comparison 2 of 12 ENVE/GEOE 224 (W2021) – P. Schmidt 1 2

2021-03-28 2 Tests On the Difference Between Two Means ( 𝜎 ଵ and 𝜎 ଶ Known, Data Normal or 𝑛 ଵ ≥ 30 and 𝑛 ଶ ≥ 30 ) 3 of 12 Test statistic: 𝑍 = ௑ ത భ ି௑ ത మ ିఋ ఙ భ మ ௡ భ ⁄ ାఙ మ మ ௡ మ ⁄ Critical Regions for Testing 𝐻 ଴ : 𝜇 ଵ − 𝜇 ଶ = 𝛿 𝐻 ஺ Reject 𝐻 ௢ if: 𝜇 ଵ − 𝜇 ଶ < 𝛿 𝑧 < −𝑧 ఈ 𝜇 ଵ − 𝜇 ଶ > 𝛿 𝑧 > 𝑧 ఈ 𝜇 ଵ − 𝜇 ଶ ≠ 𝛿 𝑧 < −𝑧 ఈ ଶ ⁄ or 𝑧 > 𝑧 ఈ ଶ ⁄ ENVE/GEOE 224 (W2021) – P. Schmidt Tests On the Difference Between Two Means ( 𝜎 ଵ and 𝜎 ଶ Unknown and Equal, Data Normal) 4 of 12 Test statistic: 𝑇 = ௑ ത భ ି௑ ത మ ିఋ ௡ భ ିଵ ௌ భ మ ା ௡ భ ିଵ ௌ మ మ ௡ భ ௡ మ ௡ భ ା௡ మ ିଶ ௡ భ ା௡ మ Degrees of Freedom: 𝜈 = 𝑛 ଵ + 𝑛 ଶ − 2 Critical Regions for Testing 𝐻 ଴ : 𝜇 ଵ − 𝜇 ଶ = 𝛿 𝐻 ஺ Reject 𝐻 ௢ if: 𝜇 ଵ − 𝜇 ଶ < 𝛿 𝑡 < −𝑡 ఈ, ఔ 𝜇 ଵ − 𝜇 ଶ > 𝛿 𝑡 > 𝑡 ఈ, ఔ 𝜇 ଵ − 𝜇 ଶ ≠ 𝛿 𝑡 < −𝑡 ఈ ଶ ⁄ , ఔ or 𝑡 > 𝑡 ఈ ଶ ⁄ , ఔ ENVE/GEOE 224 (W2021) – P. Schmidt 3 4

2021-03-28 3 Tests On the Difference Between Two Means ( 𝜎 ଵ and 𝜎 ଶ Unknown and Unequal, Data Normal) 5 of 12 Test statistic: 𝑇 = ௑ ത భ ି௑ ത మ ିఋ ௌ భ మ ௡ భ ⁄ ାௌ మ మ ௡ మ ⁄ Degrees of Freedom: 𝜈 = ௌ భ మ ௡ భ ⁄ ାௌ మ మ ௡ మ ⁄ మ ௌ భ మ ௡ భ ⁄ మ ௡ భ ିଵ ൗ ା ௌ మ మ ௡ మ ⁄ మ ௡ మ ିଵ ൗ Critical Regions for Testing 𝐻 ଴ : 𝜇 ଵ − 𝜇 ଶ = 𝛿 𝐻 ஺ Reject 𝐻 ௢ if: 𝜇 ଵ − 𝜇 ଶ < 𝛿 𝑡 < −𝑡 ఈ, ఔ 𝜇 ଵ − 𝜇 ଶ > 𝛿 𝑡 > 𝑡 ఈ, ఔ 𝜇 ଵ − 𝜇 ଶ ≠ 𝛿 𝑡 < −𝑡 ఈ ଶ ⁄ , ఔ or 𝑡 > 𝑡 ఈ ଶ ⁄ , ఔ ENVE/GEOE 224 (W2021) – P. Schmidt Tests On the Difference in a Paired Comparison ( 𝜎 Unknown and Differences Normal) 6 of 12 Test statistic: 𝑇 = ஽ ഥ ିௗ బ ௌ ೏ ௡ ⁄ Critical Regions for Testing 𝐻 ଴ : 𝜇 ஽ = 𝑑 ଴ 𝐻 ஺ Reject 𝐻 ଴ if: 𝜇 ஽ < 𝑑 ଴ 𝑡 < −𝑡 ఈ, ௡ିଵ 𝜇 ஽ > 𝑑 ଴ 𝑡 > 𝑡 ఈ, ௡ିଵ 𝜇 ஽ ≠ 𝑑 ଴ 𝑡 < −𝑡 ఈ ଶ ⁄ , ௡ିଵ or 𝑡 > 𝑡 ఈ ଶ ⁄ , ௡ିଵ ENVE/GEOE 224 (W2021) – P. Schmidt 5 6

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