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Sampling distributions, which serve as models for populations of quantities. Constants, such as parameters, characterize distribution properties. Random samples are drawn to calculate statistics, often for estimating unknown parameters. Central limit theorem (clt) and law of large numbers (lln) are key concepts. Examples include calculating probabilities of statistics from normal distributions and approximating binomial distributions.
Typology: Study notes
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Probability distribution of ] : serves as a model of a population of quantities
. 5, #^ , 1 , etc. are parameters : constants (usually unknown) which characterize properties of the distribution
] (^) " , ] (^) # , ..., ] 8 : a random sample (independent, identically distributed random variables)
Statistic : quantity calculated from ] (^) " , ] (^) # , ..., ] 8 (& possibly known parameters), usually for the purpose of estimating an unknown parameter
Examples of statistics
] œ( ] & ] & â & ] 8
sample mean
a (^) " # 8 b
W œ ] ( ] & ] ( ] & â & ] ( ] 8 ( "
" # 8
sample variance
a b
’ a^ b^ a^ b^ a^ b “
Statistics are themselves random variables with probability distributions
TRUE FACTS about ]^ (:
Ea ]( b^ œ. (^) ]( œ.
Va ]( bœ œ 8
](
] μ 8
normal (^) Π., (^) &
Variants of TRUE FACTS 1-3 for sums :
[ œ ] (^) " & ] (^) # & â & ] 8
Ea [ bœ 8.
Va [ bœ 8 5 #
[ μ normal ˆ 8 ., 85 #‰
ex. Draw 10 students at random from UI and find out their SAT-math scores. In the nation, a randomly drawn SAT- math score has a normal distribution with a mean of 500 and a standard deviation of 100. If UI students were similar to students in the nation at large, what is the probability that ](would be greater than or equal to 560?
^ œ ] ( œ ¸ œ
( Î 8
. ( ˆ 5 È ‰ Š È ‹
560 500 100/ 10
1.90; area 0.
CLT applied to the binomial distribution
Independent success/failure trials; 1 œ prob. of success
M œ (^) œ
1 if trial is a success 0 if trial is a failure
E a bM œ 1 a œ 0 † a 1 ( 1 b & 1 † 1 b
V a bM œ 1 a 1 ( 1 b ˆ^ œ a 0 ( 1 b #^ † a 1 ( 1 b & a 1 ( 1 b#† 1 ‰
] œ M (^) " & M (^) # & â & M 8 μ binomial a 8 , 1 b
So ] is a sum ; distribution of ] can be approximated by a normal a 8 1 , 8 1 a 1 ( 1 bbdistribution
Approximation is good if 8 1 5 and 8 a 1 ( 1 b 5
ex. G uess the suit of the top card in a shuffled deck. Repeat (shuffle/guess) 100 times. What is the chance of 30 or more correct guesses? 1 œ 0.25 8 œ 100
P a ] 30 b ¸ P Š ^ (^) È"!! 30 ( a.25 100 .25 ba^ a 1 (.25b b‹œ P a ^ 1.15b
œ 0.
