18 19 paired t test n erences diff d n i 1 29 1 2 2 2

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19 Paired T-Test n erences Diff D n i = = 1 ( 29 1 2 2 2 - - = n n s difference s difference S D Note: df = n-1
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20 Example. How should a manufacturer design an experiment to see if a new steel-belted radial tire lasts longer than a current model? The tests are to be run on a variety of cars with a variety of drivers. One tire of each type was installed on the rear wheel of 20 randomly selected cars. We have a pair of observations for each car. The number of miles until tire wear-out was recorded. Test to see if there is a difference at the 0.05 significance level. Null Hypothesis: Alternative Hypothesis:
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21 Car New-Dsn Exst-Dsn Diff DiffSq. 1 57 48 9 81 2 64 50 14 196 3 102 89 13 169 4 62 56 6 36 5 81 78 3 9 6 87 75 12 144 7 61 50 11 121 8 62 49 13 169 9 74 70 4 16 10 62 66 -4 16 11 100 98 2 4 12 90 86 4 16 13 83 78 5 25 14 84 90 -6 36 15 86 98 -12 144 16 62 58 4 16 17 67 58 9 81 18 40 41 -1 1 19 71 61 10 100 20 77 82 -5 25 Sums 91 1405
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22 Test Statistic: Df = 20 -1 = 19 P-value =p(t 19 > 48.15) + p(t 19 < - 48.15) = 2* p(t 19 > 48.15) Note: p(t 19 > 48.15) is less than .005 p-value < .01
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Decision: Reject H 0 Conclusion: There is enough evidence to show that the mean of the differences is not zero. (This means there is a difference in the tires.) 23
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24 SAMPLE SIZES FOR HYPOTHESIS TESTING (NOT IN BOOK)
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Return to Four Outcomes from Four Outcomes from Hypothesis Testing Hypothesis Testing H 0 is True H 0 is False Fail to Reject H 0 Correct Decision Type II Error Reject H 0 Type I Error Correct Decision probability of a type I error = α probability of a type II error = β The power of a test is the probability of rejecting H 0 given that a specific alternative is true. It is computed as 1 – β.
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Reducing the probability of one error (type I or type II) increases the probability of the other error. To decrease the probability of both types of error, increase the sample size. This also increases the power of the test. HOW LARGE OF A SAMPLE DO WE NEED?
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Sample Size for z test for One Mean One tailed alternative hypothesis: n = (z α + z β ) 2 σ 2 / δ 2 where δ is the difference between the true mean and the mean in our hypotheses ( δ = μ - μ 0 ) Two tailed alternative hypothesis: n = (z α/2 + z β ) 2 σ 2 / δ 2 Always round up the answer up to the next integer!
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Example Suppose we want to test the hypothesis H 0 : μ = 68 H 1 : μ > 68 for an alpha of .05 when σ is known to be 5. Find the sample size required if the power of our test is to be .95 when the true mean is 69.
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