1 / 12

Welcome Back

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)

dmitri
Download Presentation

Welcome Back

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related