The probability that a call to an emergency help line is answered in less than 10 seconds is 0.8.Assume that the calls are independent of each other.Using the normal approximation for binomial with a correction for continuity, the probability that at least 75 of 100 calls are answered within 10 seconds is approximately _______
A)0.8
B)0.1313
C)0.5235
D)0.9154
E)0.8687

Question

Quizplus · Accepted Answer

To approximate the probability that at least 75 out of 100 calls are answered within 10 seconds using the normal approximation to the binomial distribution, we first calculate the mean (μ) and standard deviation (σ) of the binomial distribution. For a binomial distribution, μ = np and σ = sqrt(np(1-p)), where n is the number of trials (100 calls) and p is the probability of success (0.8).μ = np = 100 * 0.8 = 80σ = sqrt(np(1-p)) = sqrt(100 * 0.8 * 0.2) = sqrt(16) = 4To find the probability of at least 75 successes, we convert this to a z-score and use the normal distribution. We apply a continuity correction by subtracting 0.5 from 75, getting 74.5, because we're looking for "at least 75" and the normal approximation is continuous.z = (X - μ) / σ = (74.5 - 80) / 4 = -5.5 / 4 = -1.375Using a z-table or calculator, the probability of z being less than -1.375 is approximately 0.0846. However, since we want at least 75 calls, we're interested in the complement of this, which is 1 - 0.0846 = 0.9154. Therefore, the probability that at least 75 of 100 calls are answered within 10 seconds is approximately 0.9154.

The Probability That a Call to an Emergency Help Line