Math 1710 Class 12 CLT Normal Arising in Dr. Allen Back Averages - PowerPoint PPT Presentation

Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist Math 1710 Class 12 CLT Normal Arising in Dr. Allen Back Averages Sep. 21, 2016

From Z-scores to Probabilities on the Calculator Math 1710 Class 12 V1c On something like a TI-84 Normal Prob on a Calc Sampling Dist 2 nd → DISTR → normalcdf ( − 10 , 1) CLT Normal Arising in Averages will give you P ( Z < 1) . Here − 10 is a substitute for −∞ , good enough for 8 decimal places. Please always compute your Z-scores if using the calculator even though the calculator has a way of bypassing this.

From Z-scores to Probabilities on the Calculator Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist On something like a TI-89: CLT Normal catalog → F3 → 2 nd alpha N → normalcdf ( − 10 , 1) Arising in Averages

Sampling Distribution Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist CLT Suppose that the chance of a ‘yes’ answer from each individual Normal in a poll is p and each of the responses of the n people Arising in Averages participating are independent.

Sampling Distribution Math 1710 Class 12 V1c Normal Prob Suppose that the chance of a ‘yes’ answer from each individual on a Calc in a poll is p and each of the responses of the n people Sampling Dist participating are independent. CLT Then if we take one such poll of size n , we will get k yes Normal Arising in responses and compute an observed proportion of yeses of Averages p = k ˆ n .

Sampling Distribution Math 1710 Class 12 Then if we take one such poll of size n , we will get k yes V1c responses and compute an observed proportion of yeses of Normal Prob on a Calc p = k Sampling Dist ˆ n . CLT Normal Arising in If we have a (typically large) population, each sample gives rise Averages to a number ˆ p . Each sample also has a certain probability. So we can view ˆ p as a random variable with a probability distribution reflecting the chance of all the different values of ˆ p showing up. This random variable is what we mean by the sampling distribution of ˆ p .

Sampling Distribution Math 1710 Class 12 V1c Normal Prob If we have a (typically large) population, each sample gives rise on a Calc to a number ˆ p . Each sample also has a certain probability. So Sampling Dist we can view ˆ p as a random variable with a probability CLT distribution reflecting the chance of all the different values of ˆ p Normal Arising in showing up. Averages This random variable is what we mean by the sampling distribution of ˆ p . It is a probability based “theoretical” idea, though we can explore it by simulation.

Sampling Distribution Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist Similarly, for quantitative data following some model, we can CLT think about all possible samples of size n , and keep track of the Normal Arising in probabilities of all the different ¯ x ′ s showing up. This is the Averages sampling distribution of ¯ x .

Sampling Distribution Math 1710 Class 12 V1c Normal Prob on a Calc One doesn’t actually need a population to do this. If one has a Sampling Dist random variable describing one individual’s response, (perhaps CLT X ∼ Bernoulli ( p ) in the polling (categorical with 2 responses) Normal Arising in case or a more general X in the quantitative case, we can just Averages consider n independent copies of X (keeping track of the n responses in a sample of size n ) to determine our summary statistic ˆ p or ¯ x .

Three CLT problems Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist CLT (1) 80% of all cars on the interstate speed. Normal Randomly sample 50. Arising in Averages What proportion of speeders might we see?

Three CLT problems Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist (1) 80% of all cars on the interstate speed. CLT Randomly sample 50. Normal What proportion of speeders might we see? Arising in Averages (e.g.: A policeman has a quota of at least 35 speeders to ticket. What is the chance he’ll meet his quota in such a sample?)

Three CLT problems Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist (2) At birth, babies average 7.8 pounds with a standard CLT deviation of 2.1 pounds. Normal Arising in 34 Babies born near a large possibly polluting factory average Averages 7.2 pounds.

Three CLT problems Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist (2) At birth, babies average 7.8 pounds with a standard CLT deviation of 2.1 pounds. Normal 34 Babies born near a large possibly polluting factory average Arising in Averages 7.2 pounds. Is that unusually low?

Three CLT problems Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist CLT (3) Suppose SAT’s have a mean of 500, standard deviation of Normal 100 and are approximately normal. Arising in Averages Form means of samples of 20 students.

Three CLT problems Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist (3) Suppose SAT’s have a mean of 500, standard deviation of CLT 100 and are approximately normal. Normal Form means of samples of 20 students. Arising in Averages What distribution do these follow? What is the chance of a sample averaging over 600? Over 550?

Proportion Case of the CLT Math 1710 Class 12 V1c Normal Prob on a Calc X ∼ Binomial(n,p) Sampling Dist X approximated by normal RV Y ∼ N ( np , √ npq ) . CLT Suppose we keep track of the observed proportion Normal Arising in Averages p = X ˆ n here.

Proportion Case of the CLT Math 1710 Class 12 V1c X ∼ Binomial(n,p) X approximated by normal RV Y ∼ N ( np , √ npq ) . Normal Prob on a Calc Suppose we keep track of the observed proportion Sampling Dist CLT p = X ˆ Normal n Arising in Averages here. Then ˆ p is an RV approximated by Y / n ∼ N ( p , SD (ˆ p )) . where � pq SD(ˆ p ) = n .

Proportion Case of the CLT Math 1710 Class 12 X ∼ Binomial(n,p) V1c X approximated by normal RV Y ∼ N ( np , √ npq ) . Normal Prob Suppose we keep track of the observed proportion on a Calc Sampling Dist p = X ˆ CLT n Normal Arising in Averages here. Then ˆ p is an RV approximated by Y / n ∼ N ( p , SD (ˆ p )) . where � pq SD(ˆ p ) = n . � pq Thus N ( p , n ) is the sampling distribution of ˆ p .

Proportion Case of the CLT Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist Some books uses the notation CLT Normal � pq Arising in SD(ˆ p ) = n . Averages

Proportion Case of the CLT Math 1710 Class 12 V1c Normal Prob on a Calc Some books uses the notation Sampling Dist CLT � pq Normal SD(ˆ p ) = n . Arising in Averages SD(ˆ p ) stands for Standard Deviation of the Sampling Distribution of ˆ p .

Proportion Case of the CLT Math 1710 Class 12 V1c Some books uses the notation Normal Prob on a Calc � pq SD(ˆ p ) = n . Sampling Dist CLT SD(ˆ p ) stands for Standard Deviation of the Sampling Normal Arising in Distribution of ˆ p . Averages And the notation � p ˆ ˆ q SE(ˆ p ) = n . SE(ˆ p ) stands for the Standard Error of ˆ p .

Proportion Case of the CLT Math 1710 Class 12 V1c Normal Prob on a Calc Sampling Dist Others, including your textbook, I believe, are happy to refer to CLT both of these as standard errors of ˆ p . One is exact (but hard to Normal Arising in know in practice) while the other is only an estimate, but Averages feasible to produce.

Quantitative Case of the CLT Math 1710 Class 12 V1c Suppose X is a random variable with mean µ and Normal Prob standard deviation σ. on a Calc Sampling Dist Let X 1 , X 2 ,. . . , X n be n independent copies of X . CLT Set Normal X = X 1 + X 2 + . . . + X n Arising in ¯ . Averages n Then 1 The mean of ¯ X is µ. σ 2 The standard deviation of ¯ X is √ n . ¯ X is approximately normal as n gets large. 3

Quantitative CLT - Why? Math 1710 Class 12 V1c Normal Prob on a Calc X = X 1 + X 2 + . . . + X n ¯ Sampling Dist . n CLT Normal Arising in 1 The mean of ¯ X is µ. Averages σ 2 The standard deviation of ¯ X is √ n . ¯ X is approximately normal as n gets large. 3

Quantitative CLT - Why? Math 1710 Class 12 X = X 1 + X 2 + . . . + X n ¯ V1c . n Normal Prob on a Calc 1 The mean of ¯ X is µ. Sampling Dist σ CLT 2 The standard deviation of ¯ X is √ n . Normal Arising in ¯ Averages X is approximately normal as n gets large. 3 (1) is immediate from E ( X 1 + X 2 + . . . + X n ) = nE ( X ) so X ) = nE ( X ) E ( ¯ = E ( X ) . n

Quantitative CLT - Why? Math 1710 Class 12 X = X 1 + X 2 + . . . + X n ¯ . V1c n Normal Prob on a Calc 1 The mean of ¯ X is µ. Sampling Dist σ 2 The standard deviation of ¯ CLT X is √ n . Normal Arising in ¯ X is approximately normal as n gets large. Averages 3 (2) is immediate from Var ( X 1 + X 2 + . . . + X n ) = nVar ( X ) so X ) = nVar ( X ) = Var ( X ) Var ( ¯ . n 2 n And so the std. dev of ¯ X is √ n . σ