According to the NYS Attorney General\'s Office, LIRR trains were late on the 6:

Question

According to the NYS Attorney General's Office, LIRR trains were late on the 6:14 Port washington train on Met's Game days 78% of the time, 60 randomly selected home game days were selected. Let r be a random variable that represents the number of 6:14 trains that are late. The clinic wants a probability distribution for r (a) Write a brief but complete description in which you explain why the normal approximation to the binomial would apply. Are the assumptions satisfied? Explain. (b) Estimate p(r s 49) (c) Estimate p(49 sr s51)

Explanation / Answer

Solution

(a) The normal distributioncan be used as an approximation to the binomial distribution, under certain circumstances, namely: If X ~ B(n, p) and if n is large then X is approximately N(np, npq) ,with the following conditions:

np > 5 and nq > 5, then the binomial random variable is approximately normally distributed with mean m =np and standard deviation s = sqrt(npq).

where n = 60 and p = 0.78

np = 46.8 and nq = 13.2 this condition is statisfied.

(b) P(r<=49) = P(r<49.5) (using continuity correction)

Mean = np = 46.8 and variance = npq = 10.296

P(z<(49.5-46.8)/10.2960.5) = P(z<0.8415) = 0.8

(c) P(49<=r<=51) = P(48.5<r<51.5)

P(0.53<z<1.46) = 0.226

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