Sampling distribution STAT 587 (Engineering) Iowa State University - PowerPoint PPT Presentation

Sampling distribution STAT 587 (Engineering) Iowa State University September 23, 2020

Sampling distribution Sampling distribution The sampling distribution of a statistic is the distribution of the statistic over different realizations of the data . Find the following sampling distributions: ind ∼ N ( µ, σ 2 ) , If Y i Y − µ Y and S/ √ n. If Y ∼ Bin ( n, p ) , Y n .

Sampling distribution Normal model Normal model ind ∼ N ( µ, σ 2 ) , then Let Y i Y ∼ N ( µ, σ 2 /n ) . Sampling distribution for N(35, 25) average n = 20 n = 30 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 density n = 40 n = 50 0.6 0.4 0.4 0.2 0.2 0.0 0.0 30.0 32.5 35.0 37.5 40.0 30.0 32.5 35.0 37.5 40.0 average

Sampling distribution Normal model Normal model ind ∼ N ( µ, σ 2 ) , then the t-statistic Let Y i T = Y − µ S/ √ n ∼ t n − 1 . Sampling distribution of the t−statistic n = 20 n = 30 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 density 0.0 0.0 n = 40 n = 50 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 −4 −2 0 2 4 −4 −2 0 2 4 t

Sampling distribution Binomial model Binomial model Let Y ∼ Bin ( n, p ) , then � Y � p = 0 , 1 n, 2 n, . . . , n − 1 P n = p = P ( Y = np ) , , 1 . n Sampling distribution for binomial proportion p = 0.5 p = 0.8 0.3 0.2 n = 10 0.1 0.0 0.100 0.075 n = 100 0.050 0.025 0.000 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Sample proportion (y/n)

Sampling distribution Approximate sampling distributions Approximate sampling distributions Recall that from the Central Limit Theorem (CLT): n � ∼ N ( nµ, nσ 2 ) ∼ N ( µ, σ 2 /n ) · X = S/n · S = X i and i =1 for independent X i with E [ X i ] = µ and V ar [ X i ] = σ 2 .

Sampling distribution Approximate sampling distributions Approximate sampling distribution for binomial proportion ind If Y = � n i =1 X i with X i ∼ Ber ( p ) , then Y � p, p [1 − p ] � ∼ N · . n n Approximate sampling distributions for binomial proportion p = 0.5 p = 0.8 3 2 n = 10 1 0 10.0 7.5 n = 100 5.0 2.5 0.0 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 Sample proportion (y/n)

Sampling distribution Summary Summary Sampling distributions: ind ∼ N ( µ, σ 2 ) , If Y i Y ∼ N ( µ, σ 2 /n ) and Y − µ S/ √ n ∼ t n − 1 . If Y ∼ Bin ( n, p ) , � Y � P n = p = P ( Y = np ) and � � p, p [1 − p ] Y ∼ N · . n n If X i independent with E [ X i ] = µ and V ar [ X i ] = σ 2 , then n � ∼ N ( nµ, nσ 2 ) S = X i · i =1 and ∼ N ( µ, σ 2 /n ) X = S/n ·