Understanding Sampling Distributions
A sampling distribution describes the probability distribution of a sample statistic (like the mean, variance, etc.) calculated from multiple random samples drawn from the same population. It helps us understand the variability of these statistics.
Analyzing Distribution Types
Let's examine the given options:
- Student’s ‘t’ distribution: This is a common sampling distribution used for the sample mean when the population standard deviation is unknown and the sample size is small. It arises from the distribution of the t-statistic.
- Fisher’s ‘F’ distribution: This is a sampling distribution typically used for the ratio of two variances. It's fundamental in ANOVA and regression analysis.
- $\chi^2$ (chi square) distribution: This is a sampling distribution that arises in several contexts, such as the distribution of the sum of squared standard normal variables or in tests related to variance.
- Poisson distribution: This is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It models count data (e.g., number of calls per hour) and is not inherently a distribution of a sample statistic like the mean or variance.
Conclusion on Sampling Distributions
While the Student’s ‘t’, Fisher’s ‘F’, and $\chi^2$ distributions are fundamentally derived as sampling distributions of specific statistics, the Poisson distribution serves as a model for count data. Therefore, the Poisson distribution is not typically classified as a sampling distribution in the same context as the others.
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 ?
