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# Introduction to Statistics STAT 2

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This 27 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 2 at University of California - Berkeley taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/226739/stat-2-university-of-california-berkeley in Statistics at University of California - Berkeley.

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Date Created: 10/22/15

STAT 2 Lecture 20 Box models and the law of averages Recall The binomial formula Pk out Of n no of ways prob of each way 14 k n k 1 kn kplt 1 Example 391 have a boX containing seven red marbles and three blue marbles I randomly draw a marble from the box then replace it I do this ten times What39s the probability I get eight or more red marbles Example P8 or more red P8 red P9 red P10 red P8 red 1082078032 02335 P9 red 10079032 01211 P10 red 0710 00282 P8 or more red 03828 38 Recall When can you use the binomial formula 0 n the number of trials must be xed in advance 39 p the probability of success on one trial doesn39t change 0 Trials must be independent Today The law of averages for proportions Generalising the law of averages Box models T he law of averages for proportions The law of averages basic form If I toss a coin a large but xed number of times the proportion of heads will be around half Assumptions PH always 12 Independence True or false I toss a fair coin ten times and get ten heads True or false the next toss is more likely to be a tail than a head False 0 With a fair coin probability of getting a head is always 12 independent of what39s happened previously True or false I toss a fair coin twenty times True or false I39m very likely to get between 45 and 55 heads False P4555 heads in 20 tosses P9 to 11 heads in 20 tosses P9 H P10 H Pll H P9 heads 209111291211 01602 P10 H 01762 Pll H 01602 P9 to 11 heads 04966 4966 True or false I toss a fair coin 1000 times True or false I39m very likely to get between 45 and 55 heads True P4555 heads in 1000 tosses P450 to 550 H in 1000 tosses P450 H P550 H Using a computer P450 to 550 heads 9986 True or false I toss a fair coin a million times True or false I39m very likely to get between 499950 heads and 500050 heads False Pbetween 499950 and 500050 H 804 Not likely Reason 499950 to 500050 heads means 4999550005 heads a very small percentage range True or false I toss a fair coin twenty times True or false I39m very unlikely to get at least 60 heads False Pat least 60 heads in 20 tosses P12 or more heads in 20 tosses P12 H P13 H 2517 Not that unlikely True or false I toss a fair coin 1000 times True or false I39m very unlikely to get at least 60 heads True Pat least 60 heads Pat least 600 heads out of 1000 l in 7 billion Fair to say this is very unlikely The law of averages variations The chance of the proportion of successes being within an interval that contains the true proportion increases as the sample size gets larger e g chance of proportion of heads being between 40 to 60 increases as of tosses increases The law of averages variations The chance of the proportion of successes being within an interval that does not contain the true proportion decreases as the sample size gets larger e g chance of proportion of heads being between 60 to 80 decreases as of tosses increases The law of averages variations The chance of the number of successes being within plus or minus a xed number of the expected number decreases as the sample size gets larger e g chance of number of heads being within 10 of the expected number decreases as n increases II General ising the law of averages The sum of die rolls 391 roll a die 60 times What do you expect the sum to be How many ones do we expect Pone 16 We expect about 10 ones and 10 twos and 10 threes The sum of die rolls If we get 10 of each number sum 0f 60 die rolls will be 101102103104 105106 210 Expect a sum of 210 in 60 rolls or an average of 35

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