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Standard error; population 12,19,13,16

Short answer

Standard error of the sample mean is the standard deviation of the sampling distribution of x̄. All six samples of size two are (12,19),(12,13),(12,16),(19,13),(19,16),(13,16); population σ=2.7386, so SE=(2.7386/√2)√(2/3)=1.5811.

Define standard error of sample mean. A population consist of the four numbers 12, 19, 13, 16. i) Write down all possible sample size of two without replacement. ii) Find standard error of the sample mean.

Standard error of the sample mean: the standard deviation of the sampling distribution of x̄. It is σ/√n for infinite population (or sampling with replacement) and (σ/√n)√((N − n)/(N − 1)) for sampling without replacement from a finite population of size N.

Population: 12, 19, 13, 16 (N = 4), n = 2 without replacement.

i) All possible samples

No.SampleSample mean
1(12, 19)15.5
2(12, 13)12.5
3(12, 16)14
4(19, 13)16
5(19, 16)17.5
6(13, 16)14.5

ii) Standard error of the sample mean

Population mean and standard deviation:

μ = Σx/N = 60/4 = 15

σ² = Σ(x − μ)²/N = (9 + 16 + 4 + 1)/4 = 30/4 = 7.5

σ = √7.5 = 2.7386

With sampling without replacement from a finite population:

SE = (σ/√n) × √((N − n)/(N − 1)) = (2.7386/√2) × √((4 − 2)/(4 − 1))

= 1.9365 × 0.8165 = 1.5811

(Direct check: the standard deviation of the 6 sample means is 1.5811.)

Mean of sample means = 15 = μ.

Answer: SE of the sample mean = 1.5811.

✓ Answer checked by Claude Sonnet 5.5

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