Formula for Standard Error of Mean Differences
The standard error of the difference between two means quantifies the expected variation in the difference between sample means if multiple pairs of samples were drawn. It is a fundamental measure used in hypothesis testing to assess the significance of the difference between two group averages.
Standard Error of Difference Between Two Means
The correct formula for the standard error of the difference between two independent sample means ($\bar{x}_1 - \bar{x}_2$), especially when population variances ($\sigma_1^2$ and $\sigma_2^2$) are known or assumed unequal, is:
In this formula, $\sigma_1^2$ and $\sigma_2^2$ represent the population variances for the two groups, and $n_1$ and $n_2$ denote their respective sample sizes. This calculation determines the standard deviation of the sampling distribution of the difference between two means.
Comparison of Formula Options
Evaluating the provided options reveals the correct choice:
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Option 1: $\frac{\sigma_p}{\sqrt{n}}$This formula represents the standard error of the mean for a single sample.
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Option 2: $\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$This is the accurate formula for calculating the standard error of the difference between two means.
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Option 3: $\frac{\sigma_1^2}{\sigma_2^2}$ when $\sigma_1^2 < \sigma_2^2$This expresses a ratio of variances, commonly used in F-tests for equality of variances, not for standard error of mean differences.
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Option 4: $\sqrt{\frac{p_1 q_1}{n_1} + \frac{p_2 q_2}{n_2}}$This formula calculates the standard error of the difference between two proportions, applicable to categorical data.
- Option 5: This option is empty and does not provide a formula.
Based on statistical principles, Option 2 correctly provides the formula for the standard error of mean differences between two sample groups.
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?
- Standard deviation of a sampling distribution of a statistic is termed as,
