Sampling Distribution Math Example 1

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Example 1

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A population has μ=50\mu=50 and σ=12\sigma=12. For random samples of size n=36n=36, describe the sampling distribution of Xˉ\bar{X}: find its mean, standard error, and shape.

Solution

  1. 1
    Mean of sampling distribution: μXˉ=μ=50\mu_{\bar{X}} = \mu = 50
  2. 2
    Standard error: σXˉ=σn=1236=126=2\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{12}{\sqrt{36}} = \frac{12}{6} = 2
  3. 3
    Shape: by CLT (n=36 ≥ 30), approximately Normal even if population is not Normal
  4. 4
    Full description: Xˉ∼N(50,2)\bar{X} \sim N(50, 2)

Answer

Xˉ∼N(μ=50, SE=2)\bar{X} \sim N(\mu=50,\ SE=2). Sample means are normally distributed around 50 with SD=2.
The sampling distribution describes the distribution of sample means across all possible samples of size n. Its mean equals the population mean (unbiased); its SD (standard error) = σ/√n. The CLT guarantees approximate normality for large n.

About Sampling Distribution

The probability distribution of a statistic (such as the sample mean) computed from all possible random samples of the same size drawn from a population.

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