List of top Statistics Questions

Let X₁, X₂ ,…, X₂₅ be a random sample from a 𝑁(μ, 1) distribution, where μ ∈ \(\R\) is unknown. Consider testing of the hypothesis H₀ : μ = 5.2 against H₁ : μ = 5.6. The null hypothesis is rejected if and only if \(\frac{1}{25}\sum^{25}_{i=1}X_i > k\) , for some constant k. If the size of the test is 0.05, then the probability of type-II error equals __________ (round off to 2 decimal places)

Five men go to a restaurant together and each of them orders a dish that is different from the dishes ordered by the other members of the group. However, the waiter serves the dishes randomly. Then the probability that exactly one of them gets the dish he ordered equals __________ (round off to 2 decimal places)

A vaccine, when it is administered to an individual, produces no side effects with probability \(\frac{4}{5}\), mild side effects with probability \(\frac{2}{15}\) and severe side effects with probability \(\frac{1}{15}\). Assume that the development of side effects is independent across individuals. The vaccine was administered to 1000 randomly selected individuals. If X₁ denotes the number of individuals who developed mild side effects and X₂ denotes the number of individuals who developed severe side effects, then the coefficient of variation of X₁ + X₂ equals __________ (round off to 2 decimal places)

Let X₁, X₂ , … , X₉ be a random sample from a N(μ, σ2) distribution, where μ ∈ \(\R\) and σ > 0 are unknown. Let the observed values of \(\overline{X}=\frac{1}{9}\sum^9_{i=1}X_i\) and \(S^2=\frac{1}{8}\sum^9_{i=1}(X_i-\overline{X})^2\) be 9.8 and 1.44, respectively. If the likelihood ratio test is used to test the hypothesis H₀ : μ = 8.8 against H₁ : μ > 8.8, then the p-value of the test equals __________ (round off to 3 decimal places)