Exercise 11.3 Let X1,X2, . . . be a family of IID random variables that follow the uniform distribution U(0, π/2). In order to evaluate the integral I =

∫ π/2 0 sin x dx, let ¯ Zn =

Σn i=1 sinXi/n. Determine the number of samples necessary to guarantee the confidence α = 95% with the confidence interval

ε = 0.1%, that is, P{| ¯ Zn − I| ≤ 0.001} = 0.95. Hint: First show that I = πE[sinX1]/2 and V[sinX1] = (π2 − 8)/2π2.