Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Random Variables & Distribution Functions
For n ≥ 2, let X1, X2, ..., Xn be a random sample from a N(μ, σ2) population, where μ ∈(-∞, ∞) and σ > 0 are unknown. Define \(\bar{X}=\frac{1}{n} \sum_{j=1}^n X_j\) and \(S^2=\frac{1}{n-1} \sum_{i=1}^n\left(X_i-\bar{X}\right)^2\). For any α ∈(0, 1) and any positive integer m, let zα denote the (1 - α)th quantile of the standard normal distribution and tm, α denote the (1 - α)th quantile of t-distribution with m degrees of freedom. Then which of the following represent 90% confidence intervals for μ ?
1
\(\left(\bar{X}-\frac{s}{\sqrt{n}} t_{n-1,0.05}, \bar{X}+\frac{s}{\sqrt{n}} t_{n-1,0.05}\right)\)
2
\(\left(\bar{X}-\frac{\sigma}{\sqrt{n}} z_{0.05}, \bar{X}+\frac{\sigma}{\sqrt{n}} z_{0.05}\right)\)
3
\(\left[\bar{X}-\frac{s}{\sqrt{n}} t_{n-1,0.9}, \infty\right)\)
4
\(\left(-\infty, \bar{X}-\frac{s}{\sqrt{n}} t_{n-1,0.9}\right)\)