Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the left of:

A) x
B) -x
C) x + 0.5
D) x - 0.1
E) -x - 0.1

Question

Given that X is a binomial random variable, the binomial probability P(X   x)  is approximated by the area under a normal curve to the left of:A)  xB)  -xC)  x + 0.5D)  x - 0.1E)  -x - 0.1

Yamna Siddiqui · Accepted Answer

Final Answer : C, Explanation :  The normal distribution is an approximation to the binomial distribution when n is large and p is not too close to 0 or 1. The mean of the normal distribution is given by μ=np and the standard deviation is given by σ=$\sqrt{np(1-p)}$. To use the normal approximation, we need to standardize the binomial random variable as follows: z=(X-μ)/σ. In this case, we want to find the binomial probability P(X<x), which is equivalent to finding the probability that z<(x-μ)/σ. Since we are approximating the binomial distribution with a normal distribution, we know that the area under the normal curve to the left of z represents the probability of interest. Therefore, we need to find the value of z that corresponds to x in our standardized normal distribution. We can do this by solving the equation (x-μ)/σ=z for z, which gives z=(x-μ)/σ=(x-np)/$\sqrt{np(1-p)}$. The best approximation for the binomial probability will be obtained using z=(x+0.5-np)/$\sqrt{np(1-p)}$, which corresponds to the midpoint correction of the continuity correction. Therefore, the answer is C.

Given that X is a binomial random variable, the binomial probability P(X x) is approximated by the area under a normal curve to the left of:

A) x
B) -x
C) x + 0.5
D) x - 0.1
E) -x - 0.1

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