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STAT140 – Section 1

STAT140 – Section 1. Bronwyn Loong bloong@fas.harvard.edu Mon 9 th Feb 2009. Section 1 – Practice Problem.

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STAT140 – Section 1

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  1. STAT140 – Section 1 Bronwyn Loong bloong@fas.harvard.edu Mon 9th Feb 2009

  2. Section 1 – Practice Problem An investigator wants to test whether Gatorade brand sports drink does a better or worse job than water of keeping runners hydrated while exercising. His proposed experimental design is as follows: • Experimental unit : athletes from Harvard college athletics team • Sample size : 20 • Treatment : 10 units Gatorade; 10 units water (by random assignment mechanism) • Experiment setup: • athletes complete a 5 min warm –up then run 1 hour • Athlete run for one hour and are hydrated with assigned treatment at 10 minute intervals • After the full hour of running, each athlete is clocked for a 1 mile run • Response variable of interest: time to run 1 mile • Proposed statistical test: unpaired two-sample t-test to for difference in mean time to run 1 mile between Gatorade and water groups COMMENT ON EACH ELEMENT OF THE EXPERIMENTAL DESIGN

  3. Section 1 – Practice Problem • To improve the experiment, the following modifications are made : • Experimental unit: male athletes from Harvard college cross country team • Treatment: 10 units Gatorade; 10 units water (by random assignment mechanism, then groups checked for balance across height and weight, running performance (5 fastest times) • Experiment setup • Athletes complete a 5 minute warm-up • Each athlete completes and is clocked for a 1mile run • Athletes run for one hour and are hydrated with specific amounts of Gatorade/water at specific intervals • After the first mile plus a full hour, each runner is clocked for another mile run. • Response variable of interest: % increase in time it took the athlete to run 1 mile between first and second run • Proposed statistical test: unpaired two-sample t-test to for difference in mean % increase in time to run 1 mile between Gatorade and water groups

  4. Gatorade experiment results

  5. Questions • Why was it necessary for all runners to run a mile before being hydrated? • What assumptions do you need to check to conduct the statistical test? • Write down the statistical model. • Conduct the statistical test

  6. Check for equal variances > var.test(x=time[treat=="W"],y=time[treat=="G"],ratio=1,alternative="two.sided",conf.level=0.95) F test to compare two variances data: time[treat == "G"] and time[treat == "W"] F = 1.7451, num df = 9, denomdf = 9, p-value = 0.4195 alternative hypothesis: true ratio of variances is not equal to 1 95 percent confidence interval: 0.4334578 7.0257572 sample estimates: ratio of variances 1.745099

  7. Two-sample t-test results > t.test(time~treat, data=gatorade.data, + alternative = "two.sided", + mu = 0, paired = FALSE, var.equal = TRUE, + conf.level = 0.95) Two Sample t-test data: time by treat t = -1.1462, df = 18, p-value = 0.2667 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -10.283661 3.023661 sample estimates: mean in group G mean in group W 6.70 10.33

  8. Improving the experiment • How would you design a study to conduct a paired t-test? • Actually the observations were taken from only 10 runners, and each runner was asked to perform under both conditions, Gatorade and water, a few days apart (randomised which liquid was used first). These results are below. • Write down the statistical model and conduct a hypothesis test for the paired data.

  9. Paired t-test results > t.test(time~treat, data=gatorade.data, + alternative = "two.sided", + mu = 0, paired = TRUE, var.equal = TRUE, + conf.level = 0.95) Paired t-test data: time by treat t = -2.696, df = 9, p-value = 0.02455 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -6.6758583 -0.5841417 sample estimates: mean of the differences -3.63

  10. Problem 2 • (Box and Hunter Ch 3 Problem 9 pg 127) An agricultural engineer obtained the following data on two methods of drying corn and asked for statistical analysis. What questions would you ask him? Under what circumstances would you be justified in analysing the data using a) a paired t-test b) an unpaired t-test or c) something else??

  11. Some notes on Statistical Inference • Unbiased estimators Let be a point estimator of a parameter θ. Then is an unbiased estimator if E[ ] = θ. In words, the mean of the distribution of the estimator is equal to the parameter estimated Some common unbiased point estimators

  12. Some notes on Statistical Inference • Unbiased estimators – example – show s2 is an unbiased estimator of σ2 Therefore

  13. Some notes on Statistical Inference • Consistency Ideally, we would like the estimator to get closer to the quantity being estimated as the amount of information in the sample (ie sample size) increases. Definition The estimator is said to be a consistent estimator of θ, if for any positive number ε In words we write converges in probability to θ. Example. – by law of large numbers is a consistent estimator of µ

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