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Chapter 5: Problem 6

Let \(Y\) be \(b\left(400, \frac{1}{5}\right)\). Compute an approximate value of \(P(0.25

Short Answer

Expert verified
The approximate probability of \(P(Y > 100)\) can be found by computing the complement of the CDF. This involves practical use of a binomial distribution calculator or statistical software.

Step by step solution

01

Solve for Y

Begin by solving for \(Y\) in the inequality \(Y / 400 > 1/4\). Multiply both sides by 400 to yield \(Y > 400/4\), so \(Y > 100\).
02

Compute the CDF

Next, compute the CDF of \(Y\) at \(Y = 100\). Recall that the CDF of a binomial random variable at \(\alpha\) is given by \(\sum_{k=0}^{\alpha} {n \choose k} (p^k) (1-p)^{n-k}\). Therefore, substitute \(\alpha = 100\), \(n = 400\), and \(p = 1/5\) into this formula and compute the resulting sum. Because calculating this sum directly may be infeasible due to the large number of terms, consider using a binomial distribution calculator or statistical software.
03

Compute the Complement of the CDF

Finally, compute the complement of the CDF by subtracting the CDF from 1. This yields the probability that \(Y > 100\), which is the event of interest. Because CDFs represent probabilities of the form \(less than or equal to\), its complement represents the probability of the form \(greater than\).

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