Solve both questions hand written and show work

1. It is known that a sample consisting of the values 2, 5, 2, 4, and 3 comes from a discrete random variable with the probability mass function Px(x) p2(1 -p) *-1 Find the maximum likelihood estimate (MLE) of p (15 points). 2. The ultimate tensile strength of steel is being tested by a mechanical engineer. He testes 42 specimens and obtains sample mean and variance are 420 MPa and 100 MPa, respectively. (a) What is the 1-a =90% two-side confidence interval of the mean ultimate tensile strength? (10 points) (b) How many additional tests are needed to cut the confidence interval width in part (a) by half? (10 points)