Reference no: EM132394803

A manufacturing processes produces tablets with on average μ units of active ingrediant. The company making the tablets wants the actual amount of active ingrediant to be within 1 unit of μ at least 75% of the time. Use Chebyshev's inequality to find the largest standard deviation σ that can be tolerated. 

The Normal distribution is often used to approximate the binomial distribution for large n. It is also a good approximation for relatively small values of n as long as p is not too small or too large. Let n = 25 and p = .4

(a) Suppose X is a binomial random variable with parameters n and p as above. Compute the probability p(X = 8).

(b) Suppose Y is a Normal random variable with μ = np and σ2 = np(1 - p). Compute the probability that p(7.5 ≤ Y ≤ 8.5).

The "Rule of Thumb" is that the Normal approximation is good enough if p ± 3 pq is between 0 and 1.

n

(c) Show that p ± 3 pq is bewtween 0 and 1 if and only if n

n > 9(p) and n > 9(q). qp

Moral: The Normal approximation is considered good enough if n>9( max{p,q})

min{p, q}

(d) How large should n be to approximate the binomial distribution with p = .5? .8? .99? .999?