Teaching CSIR NET Mock Test Series Mathematical Science Statistics & Exploratory Data Analysis Random Variables & Distribution Functions
Let X1....X12 be a random sample from the N(2, 4) distribution and Y1...Y15 be a random sample from the N(-2, 5) distribution, where N(μ, σ2) denotes a normal distribution with mean μ and variance σ2. Assume that the two random samples are mutually independent. Let \(\rm \bar X=\frac{1}{12}\Sigma_{i=1}^{12}X_i, S_1^2=\frac{1}{11}\Sigma_{i=1}^{12}(X_i-\bar X)^2,\) \(\rm \bar Y=\frac{1}{15}\Sigma_{j=1}^{15}Y_i, S_2^2=\frac{1}{14}\Sigma_{j=1}^{15}(Y_i-\bar Y)^2\)
Which of the following statements are true?
1
The distribution of X̅ + Y̅ is \(\rm N\left(0, \frac{2}{3}\right)\)
2
The distribution of \(\rm \frac{1}{20}(55S_1^2+56S_2^2)\ is \ \chi_{26}^2\)
3
The distribution of \(\rm \frac{5}{4}\frac{S_1^2}{S_2^2}\) is F11, 14
4
The distribution of \(\rm \frac{2\sqrt3(\bar Y+2)}{S_1}\) is t14