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

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 4 12 independent random samples of each of two alloys were obtained to compare their strengths. Alloy A had a mean strength of 31,400 kN/m 2 with a coefficient of variation of 23%; the corresponding values for Alloy B were 27,100 kN/m 2 and 12%. Do the mean alloy strengths differ at the 1% significance level? Example 5.4 (Revisited) 7 of 12 ENVE/GEOE 224 (W2021) – P. Schmidt Example 5.4 (Revisited) 8 of 12 ENVE/GEOE 224 (W2021) – P. Schmidt m 1 – m 2 = 0 𝑿 ഥ 𝟏 − 𝑿 ഥ 𝟐 t calc t crit 0 1.881 2.947 T t crit 7 8

2021-03-28 5 A taxi company manager is trying to decide whether the use of radial tires instead of regular belted tires improves fuel economy. Twelve cars were equipped with radial tires and driven over a prescribed test course. Without changing drivers, the same cars were then equipped with regular belted tires and driven once again over the test course. The gasoline consumption, in kilometers per liter, was recorded as follows: 1 2 3 4 5 6 7 8 9 10 11 12 Radial Tires 4.2 4.7 6.6 7.0 6.7 4.5 5.7 6.0 7.4 4.9 6.1 5.2 Belted Tires 4.1 4.9 6.2 6.9 6.8 4.4 5.7 5.8 6.9 4.7 6.0 4.9 d = X R – X B 0.1 -0.2 0.4 0.1 -0.1 0.1 0.0 0.2 0.5 0.2 0.1 0.3 Can we conclude, using a P-value, that cars equipped with radial tires give better fuel economy than those equipped with belted tires? Example 5.5 9 of 12 ENVE/GEOE 224 (W2021) – P. Schmidt Example 5.5 10 of 12 𝑫 ഥ m D = 0 t crit = t 0.05,11 t calc 0 1.796 2.485 T ENVE/GEOE 224 (W2021) – P. Schmidt 9 10