Let X_1,..., X_n be an independent trials process where each X_i has the followi
ID: 3158624 • Letter: L
Question
Let X_1,..., X_n be an independent trials process where each X_i has the following distribution: P(X_i = 1) = p, and P(X_i = 0) = 1 - p. Let us find out how many trials are needed to get 99 % confidence intervals for the actual mean with lengths less than 0.231. What is E(X_i)? (your answer should be a formula with p) What is V(X_i)? (your answer should be a formula with p) Let us find a bound on V(X_i) using calculus (the first derivative test). Let f(p) be a function whose formula is your answer to what V(X_i) is. The function f(p) has a critical point at p = (give an exact answer) so the maximum value (absolute maximum or global maximum) of f(p) = V(X_i) is (give an exact answer) In other words, V(X_i) is less than or equal to your answer here, which in turn means squareroot V(X_i) is less than or equal to (give an exact answer) Using this information, how many trials n are needed so that the 99 % confidence interval for the actual mean has a length that is less than 0.231? (round up to the next highest integer)Explanation / Answer
E(Xi)=p
Var(Xi)=1-p
for p=1/2,Var(Xi) attains its maximum value which is 1/4
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