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# 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.
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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