Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line...
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Transcript of Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line...
![Page 1: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/1.jpg)
Stat 321 – Lecture 19Central Limit Theorem
![Page 2: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/2.jpg)
Reminders
HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm
Ch. 5 “reading guide” in Blackboard Ignore page numbers
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Definitions
A statistic is any quantity whose value can be calculated from sample data.
A simple random sample of size n gives every sample of size n the same probability of occurring. Consequently, the Xi are independent random variables and every Xi has the same probability distribution.
As a function of random variables, a statistic is also a random variable and has its own probability distribution called a sampling distribution.
When n is small, we can derive the sampling distribution exactly. In other cases, we can use simulation to investigate properties of the sampling distribution.
A statistic is an unbiased estimator if E(statistic) = parameter.
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Previously
Rules for Expected ValueE(X+Y) = E(X) + E(Y)
Rules for VarianceV(X+Y) = V(X) + V(Y) IF X and Y are independent
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![Page 6: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/6.jpg)
Moral
It is often possible to find the distribution of combinations of random variables like sums and averages
What about the sample mean…
![Page 7: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/7.jpg)
The Central Limit Theorem Let X1, …, Xn be independent and identically
distributed random variables, each with mean and variance 2. Then if n is sufficiently large, has (approximately) a normal distribution with E( ) = and V( ) = 2/n. X X
X
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Example
Ethan Allen October 5, 2005
Are several explanations, could excess passenger weight be one?
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Weights of Americans
CDC: mean = 167 lbs, SD = 35 lbs Want P(T > 7500) for a random sample of
n=47 passengers Equivalent to P(X>159.57)
Sampling distribution should be normal with mean 167 lbs and standard deviation 5.11 lbs
Z = (159.57-167)/5.11 = -1.45 92.6% of boats were overweight…
![Page 10: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/10.jpg)
![Page 11: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/11.jpg)
Roulette
Total winnings vs. average winnings Find P(X > 0) Exact sampling distribution with n = 2
-1 0 1.27699 .4983 .2244
Exact sampling distribution with n =3
-1 -1/3 1/3 1.1458 .3963 .3543 .1063
![Page 12: Stat 321 – Lecture 19 Central Limit Theorem. Reminders HW 6 due tomorrow Exam solutions on-line Today’s office hours: 1-3pm Ch. 5 “reading guide” in Blackboard.](https://reader036.fdocuments.us/reader036/viewer/2022062714/56649d575503460f94a35818/html5/thumbnails/12.jpg)
Empirical Sampling Distributions Starts to get very cumbersome to do this for
large n so will use simulation instead
Approximately 35% of samples have a positive sample mean
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Number Bet
y p(y)
-$1 .9737
$35 .0263
E(Y) = -.0526
SD(Y) = 5.76
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Number bet
What does CLT predict for n = 50 spins? Approximately 47% of samples have positive
average?
Only 36%
Increases to 49% with large n?
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1000 spins
About 5% positive About 38% positive