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Welcome Back. Homework Due next Tuesday in class (on the web…) We’re in chapter 8…. n >30 or so for means, np and n(1-p) both > 5 for proportions. Review: Large Sample Confidence Intervals. 1- a confidence interval for a mean: x +/- z a /2 s/sqrt(n)

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Welcome Back

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  1. Welcome Back • Homework Due next Tuesday in class(on the web…) • We’re in chapter 8…

  2. n >30 or so for means, np and n(1-p) both > 5 for proportions Review:Large Sample Confidence Intervals • 1-a confidence interval for a mean: x +/- za/2 s/sqrt(n) • 1-a confidence interval for a proportion: p +/- za/2 p(1-p)/sqrt(n) • 1-a confidence interval for the difference between two means: x1 – x2 +/- za/2 sqrt(s21/n1+s22/n2)

  3. In General: ) ( Estimate (that is normally distributed via the CentralLimit Theorem) standard deviation Za/2of estimate +/- This gives an interval: (Lower Bound , Upper Bound) Interpretation: This is a plausible range for the true value of the number that we’re estimating. a is a tuning parameter for level of plausibility: smaller a = more conservative estimate.

  4. np and n(1-p) > 5 for all p’s… Large Sample Confidence Intervals • 1-a confidence interval for difference between two proportions: p1-p2 +/- za/2 sqrt[(p1(1-p1)/n1)+(p2(1-p2)/n2)]

  5. Designing an Experiment and Choosing a Sample Size • Example: Compare the shrinkage in a tumor due to a “new” cancer treatment relative to standard treatment • 100 patients randomly assigned to “new” treatment or standard treatment xinew = reduction in tumor size for person i under new treatment xjstd = reduction in tumor size for person j under std treatment xnew and s2new xstd and s2std Mean and sample variance of the changes in size for the new and standard treatments

  6. Suppose the data are: xnew = 25.3 snew = 2.0 xstd = 24.8 sstd = 2.3 95% Confidence Interval for difference: x1 – x2 +/- za/2 sqrt(s21/n1+s22/n2) = 0.5 +/- 0.84 What can we conclude?

  7. There’s no difference? • Can’t see a difference? • There’s a difference, but it’s too small to care about?

  8. There is a difference between: • Can’t see a difference • There’s no difference Situation for Cancer example (In cancer experiment, we can assume we care about small differences.)

  9. Can’t see a difference (that is big enough to care about) = wasted experiment • AVOID / PREVENT THE WASTE AND ASSOCIATED TEARS USE SAMPLE SIZE PLANNING

  10. Sample Size Planning • Length of a 1-a level confidence interval is: “2 za/2 std deviation of estimate” 2za/2s/sqrt(n) 2za/2p(1-p)/sqrt(n) 2za/2sqrt((s21/n1)+(s22/n2)) 2za/2sqrt[(p1(1-p1)/n1)+(p2(1-p2)/n2)]

  11. Suppose we want a 95% confidence interval no wider than W units. • a is fixed. Assume a value for the standard deviation (or variance) of the estimator. • Solve for an n (or n1 and n2) so that the width is less than W units. • When there are two sample sizes (n1 and n2), we often assume that n1 = n2.

  12. Cancer example • Let W = 0.1. Want 95% CI for difference between means with width less than W. • Suppose s2new = s2std= 6 (conservative guess) W > 2za/2sqrt((s2new/n1)+(s2std/n2)) 0.1 > 2(1.96)sqrt(6/n + 6/n) 0.1 > 3.92sqrt(12/n) 0.01 > (3.922)12/n n > 18439.68 (each group…) Book’s B = (our W)/2

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