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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: If Yi

ind

∼ N(µ, σ2), Y and Y − µ S/√n. If Y ∼ Bin(n, p), Y n .

Sampling distribution Normal model

Normal model

Let Yi

ind

∼ N(µ, σ2), then Y ∼ N(µ, σ2/n).

n = 40 n = 50 n = 20 n = 30 30.0 32.5 35.0 37.5 40.0 30.0 32.5 35.0 37.5 40.0 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4 0.6 0.0 0.1 0.2 0.3 0.4 0.0 0.2 0.4

average density

Sampling distribution for N(35, 25) average

Sampling distribution Normal model

Normal model

Let Yi

ind

∼ N(µ, σ2), then the t-statistic T = Y − µ S/√n ∼ tn−1.

n = 40 n = 50 n = 20 n = 30 −4 −2 2 4 −4 −2 2 4 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4 0.0 0.1 0.2 0.3 0.4

t density

Sampling distribution of the t−statistic

Sampling distribution Binomial model

Binomial model

Let Y ∼ Bin(n, p), then P Y n = p

• = P(Y = np),

p = 0, 1 n, 2 n, . . . , n − 1 n , 1.

p = 0.5 p = 0.8 n = 10 n = 100 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.0 0.1 0.2 0.3 0.000 0.025 0.050 0.075 0.100

Sample proportion (y/n)

Sampling distribution for binomial proportion

Sampling distribution Approximate sampling distributions

Approximate sampling distributions

Recall that from the Central Limit Theorem (CLT): S =

n

• i=1

Xi

·

∼ N(nµ, nσ2) and X = S/n · ∼ N(µ, σ2/n) for independent Xi with E[Xi] = µ and V ar[Xi] = σ2.

Sampling distribution Approximate sampling distributions

Approximate sampling distribution for binomial proportion

If Y = n

i=1 Xi with Xi ind

∼ Ber(p), then Y n

·

∼ N

• p, p[1 − p]

n

• .

p = 0.5 p = 0.8 n = 10 n = 100 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 1 2 3 0.0 2.5 5.0 7.5 10.0

Sample proportion (y/n)

Approximate sampling distributions for binomial proportion

Sampling distribution Summary

Summary

Sampling distributions:

If Yi

ind

∼ N(µ, σ2), Y ∼ N(µ, σ2/n) and

Y −µ S/√n ∼ tn−1.

If Y ∼ Bin(n, p), P Y

n = p

• = P(Y = np) and

Y n ·

∼ N

• p, p[1−p]

n

• .

If Xi independent with E[Xi] = µ and V ar[Xi] = σ2, then S =

n

• i=1

Xi

·

∼ N(nµ, nσ2) and X = S/n

·

∼ N(µ, σ2/n)