Question 1. Are the following statements true or false? Justify your answer. 1. If the moon is larger than the earth then the earth is larger than the sun. 2. The moon is larger than the earth if and only if ∃x (2x = 5) (given that the domain of x is the set of all integers). Question 2. 1. How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 = 100, where x1, x2, x3, x4, x5 are nonnegative integers. 2. Let a be a real number other than 1. By the mathematical induction, prove that 1 +a+a 2 +· · ·+a n = (a n+1 − 1)/(a − 1) for all n ≥ 1.

An automobile assembly line operation has a scheduled mean completion time, μ, of 12 minutes. the standard deviation of completion times is 1.6 minutes. it is claimed that, under new management, the mean completion time has decreased. to test this claim, a random sample of 33 completion times under new management was taken. the sample had a mean of 11.2 minutes. assume that the population is normally distributed. can we support, at the 0.05 level of significance, the claim that the mean completion time has decreased under new management? assume that the standard deviation of completion times has not changed.