II. Essay (30%)
"Eye contact is a social signal." Do you agree or disagree with this statement? Write an essay of about 300 words to explain your reasons. Provide at least three examples to support your claim.

(d) Find a 95% confidence interval of the length of a spring under a load of 1.4lb. The 95% confidence interval of the length of a spring under a load of 1.4lb is__(P)__.

(a) Find the value of k. (5%)

(b) Find the probability that rocks of types 1 and 2 together account for at most 50% of the sample. (5%)

10. Let X₁, X₂, X₃, X₄ be independent random variables that have normal distributions N(μᵢ, σ²),i = 1, 2, 3, 4. μi and σ² are the mean and variance of Xi. In order to test, at the α = 0.05 significancelevel, the hypothesis H₀: μ₁ = μ₂ = μ₃ = μ₄ = μ against all alternatives on the basis of a randomsample of size nᵢ = 3 from each of the four distributions. Determine whether we accept or reject H₀ ifthe observed values from the four distributions are, respectively, as follows:

(a) Find the mean value and variance of Y₁ = X₁ + X₂.Answer: Mean = __( A )__ ； Variance =__( B )__

(b) Find the mean value and variance of of Y₂ = 2X₁.Answer: Mean = __(C)__ ； Variance =__(D)__

(a) The (marginal) probability mass function of X, pₓ(x) =__(E)__.

(b) The (marginal) probability mass function of Y, pᵧ(y) =__(F)__.

(c) Are X and Y independent? Answer=__(G)__.

(a) The expected value of X, E(X) = __(H)__.