#--------------------------------------------------------- # File: MIT18_05S22_in-class6b-script.txt # Author: Jeremy Orloff # # MIT OpenCourseWare: https://ocw.mit.edu # 18.05 Introduction to Probability and Statistics # Spring 2022 # For information about citing these materials or our Terms of Use, visit: # https://ocw.mit.edu/terms. # #--------------------------------------------------------- class 6b CLT Jerry Slide 2 (2 minutes) CLT super important, since many phenomena are the sum or average of my We should have time to introduce joint distributions from class 7. Slide 3 Standard deviation of an average (3 minutes) Just a reminder: this is from last Thursday Slide 4 Standardization definition (2 min) Go over the points: Z has mean 0 and std dev 1: This is in the next table question Dimensionless: enough to note X and sigma have same dimension Slide 5 Table question: standardization (Work: 5 min, discuss 5 min) For repetition's sake: give the definition again How much detail in discussion depends on how the tables did. Slide 6 CLT statement (5 min) Go slowly. Emphasize that in X_n-bar approx N(mu, sigma^2/n), mu and sigma^2/n are precisely the mean and variance of X_n-bar, likewise for S_n and Z_n Slide 7 CLT statement summary (1 minute) Slides 8-11 Pictures of CLT (3 minutes) (As close as we'll come to a proof) Note: exponential distr. is asymmetric -- takes longer to converge Jen Slide 12 CQ Normal distributions (5 minutes) Point is to think in terms of standard deviations for normal distr. (Has pause) (a) Softball - answer is in the figure (b) Slightly harder Slide 13 Board question CLT (work: 10 minutes work, discussion: 8 minutes) (a) We should make sure they all get this during work --skip discussion (b) In discussion just present a model solution so they see how to organize and present such problems. Define test stat Xbar = fraction who support the team Identify probability asked for P(Xbar > .55) Identify individual X_i, with their mean (0.5) and variance (0.25) For test stat Xbar give mean (0.5) and variance (0.25/400) Standardize and use CLT: ... P(Z > 2) = 0.025 (c) Do the same but a little faster Slide 14 Table question: using the clt to approximate N(0,1) (Work: 6 minutes: discussion: 5 minutes) FOR DISCUSSION: Show figure on slide 15 Slide 15 Figure for solution to table question Slides 16, 17 Continuity correction (5 minutes) Don't belabor. In R, certain tests that rely on the CLT will are an argument to set the continuity correction to true or false. Slide 18 Bonus question SKIP