Understanding Sampling Distribution Standard Deviation
The standard deviation of a sampling distribution of a statistic measures the variability or spread of that statistic across different possible samples drawn from the same population. It quantifies how much the sample statistic is likely to differ from the true population parameter.
Defining the Term
This specific measure, the standard deviation of the sampling distribution of a statistic, has a distinct name in statistics:
- Standard Variance: Refers to the square of the standard deviation, typically associated with a population or sample, not a sampling distribution.
- Sampling Variance: Similar to standard variance, it relates to the variance within samples or populations, not the variability of a statistic itself across samples.
- Sampling Error: This is the difference between a sample statistic and the population parameter it estimates. While related, it's the difference itself, not the standard deviation *of* those differences.
- Standard Error: This is precisely the term used for the standard deviation of the sampling distribution of a statistic. It measures the precision of the statistic as an estimate of the population parameter.
Conclusion
Therefore, the standard deviation of a sampling distribution of a statistic is correctly termed the Standard error.
Important Questions from Sampling Distributions
Let X1, X2, ..., X6 be a random sample from a gamma distribution with the probability density function
\(f(x \mid \lambda)=\left\{\begin{array}{cl} \frac{\lambda^4}{6} e^{-\lambda x} x^3, & \text { if } x>0 \\ 0, & \text { if } x \leq 0 \end{array},\right.\)
where λ > 0 is unknown. Let \(T=\sum_{i=1}^6 X_i\) and ψ be the uniformly most powerful test of size α = 0.05 for testing null hypothesis H0 : λ = 1 against alternative hypothesis H1 : λ > 1. For any positive integer v, let \(\chi_{v, α}^2\) denote the (1 - α)th quantile of \(\chi_v^2\) distribution. Then the test ψ rejects H0 if and only if
For n ≥ 2, let X1, X2, ..., Xn be a random sample from a distribution with the probability density function
\(f(x \mid θ)=\left\{\begin{array}{cc} θ x^{θ-1}, & 0<x<1 \\ 0, & \text { otherwise } \end{array},\right.\)
where θ > 0 is an unknown parameter. Then which of the following is the uniformly minimum variance unbiased estimator for \(\frac{1}{\theta}\) ?
Let X1, ..., Xn be a random sample from N(μ, 1) distribution, where μ ∈ ℝ is unknown. In order to test H0 : μ = μ0 against H1 : μ > μ0, where μ0 ∈ ℝ is some specified constant, consider the following two tests:
(A) Reject H0 if and only if X̅n > c1, where c1 is such that \(P_{μ_0}\) (X̅n > c1) = α ∈ (0, 1) and X̅n = \(\frac{1}{n} \sum_{i=1}^n X_i\).
(B) Reject H0 if and only if Median {X1, ..., Xn} > c2, where c2 is such that \(P_{μ_0}\)(Median{X1, ..., Xn} > c2) = α ∈ (0, 1).
Then which of the following statements are true?
Let X1, X2, ..., Xn be a random sample from an unknown distribution with absolutely continuous cumulative distribution function (cdf) F. Let F0 be a specified absolutely continuous cdf. For testing H0 : F(x) = F0(x) for all x against H1 : F(x) ≠ F0(x) for some x, consider the following two test statistics:
\(\displaystyle T_{1, n}=\sup _{x \in \mathbb{R}}\left|\frac{1}{n} \sum_{i=1}^n I_{\left\{X_i \leq x\right\}}-F_0(x)\right| \), and \(\displaystyle T_{2, n}=\sup _{x \in \mathbb{R}} n\left|\frac{1}{n} \sum_{i=1}^n I_{\left\{X_i \leq x\right\}}-F_0(x)\right|\), where \(I_{\left\{X_i \leq x\right\}}=\left\{\begin{array}{ll}1, & \text { if } X_i \leq x \\ 0, & \text { if } X_i>x\end{array}\right.\) for i = 1, 2, ..., n.
Then which of the following statements are true?
- Which one of the following is the formula to calculate standard error of mean differences of two sample groups ?
