15 Hypothesis Testing Part 1

Cadillac example revisited 23 5 36 30 50 25 n x error

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Cadillac example revisited 23 0 . 5 36 30 50 25 . 0 0 - = - = - = - = n x error std x z x σ μ μ
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Hi-At Hotels recently instituted a fuel efficiency target of 20 mpg for its fleet of limousines. The fleet manager wants to report to top management about progress toward meeting the goal. A sample of 55 cars was selected. Following are the results. Does the sample data suggest that Hi-At is meeting its fuel efficiency goals? You want to show that fuel efficiency is more than 20 mpg. Use α = 0.05 Example: Hi-At fleet fuel efficiency 24 55 , 5 , 22 = = = n s x
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Want to show µ > 20; α = 0.05 1. Hypotheses: H0: µ ≤ 20 HA: µ > 20 2. Data: Sample x̅ = 22 s = 5 n = 55 3. Test statistic: extrem e 0 T t = 2.967 µ 0 = 20 x̅ = 22 5. Conclusion: p-value = 0.0022 < 0.05 Reject the null; conclude that fleet fuel efficiency is greater than 20 mpg. Example: Hi-At fleet fuel efficiency (p-value) 4. P-value = tdist(2.967,54,1) = 0.0022 P-value = 0.0022 25 967 . 2 55 5 20 22 . 0 0 = - = - = - = n s x error std x t x μ μ
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Consumer Watchdog Group (CWG) has accused Cheap-O of making false claims about its AA batteries. Cheap-O claims the batteries last at least 100 hours. CWG hires you to conduct a test. You take a sample of 75 batteries and install them in flashlights, radios, etc. and measure how long they last. Following are results. Does the sample data suggest that Cheap-O’s claim is inaccurate? You want to show that battery life is less than 100 hours. Use α=0.05 Example: Cheap-O batteries 26 75 , 14 , 92 = = = n s x
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Want to show µ < 100; α = 0.05 1. Hypotheses: H0: µ ≥ 100 HA: µ < 100 2. Data: Sample x̅ = 92 s = 14 n = 75 3. Test statistic: extrem e 0 T t = -4.949 µ 0 = 100 x̅ = 92 5. Conclusion: p-value = 0.000002 < 0.05 Reject the null in favor of the alternative: conclude that the battery life for Cheap-O batteries is less than 100 hours. Example: Cheap-O batteries (p-value) 4. P-value = tdist(4.949,74,1) = 0.0000 P-value = 0.000002 27 949 . 4 75 14 100 92 . 0 0 - = - = - = - = n s x error std x t x μ μ
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A Cut Above Hearts & Arrows diamond is a perfectly cut round brilliant diamond, which in addition to being ideal in proportion, is totally symmetrical and perfectly polished. To achieve the ideal reflections, the table (ratio of the top to the diameter) should be set at 55%. A recent sample of 84 diamonds was selected and resulted in the following descriptive statistics. Do you believe that Hearts & Arrows diamonds are close to ideal? IDEAL cut: Table 55% IDEAL reflections Want to show that ratio is different from ideal (55%) Example: A Cut Above 28 84 %, 4 %, 7 . 55 = = = n s x
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Want to show µ ≠ 55; α = 0.05 1. Hypotheses: H0: µ = 55 HA: µ ≠ 55 2. Data: Sample x̅ = 55.7 s = 4 n = 84 3. Test statistic: extrem e 0 T t = 1.604 µ 0 = 55x̅ = 55.7 Example: A Cut Above (p-value) extrem e 5. Conclusion: p-value = 0.1125 < 0.05 Fail to reject the null: conclude that the evidence is insufficient to show that the mean table is different from 55%. 4. P-value = tdist(1.604,83,2) = 0.1125 P-value=0.1125 29 604 . 1 84 4 55 7 . 55 . 0 0 = - = - = - = n s x error std x t x μ μ
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