Question

Consider the probability that greater than 100 100 out of 155 155 people have not been in a car accident. Assume the probability that a given person has not been in a car accident is 58% 58 % . Approximate the probability using the normal distribution. Round your answer to four decimal places.

Answer 1

The probability that more than 100 out of 155 people have not been in a car accident, assuming a 58% individual probability, using the normal distribution approximation, is approximately 0.0001.

Step-by-step explanation:

To solve this using the normal approximation, first, find the mean (μ) and standard deviation (σ) of the binomial distribution using the formula:

μ = n * p = 155 * 0.58 = 89.9

σ = sqrt(n * p * q) = sqrt(155 * 0.58 * 0.42) ≈ 6.70

Next, apply the normal approximation using the continuity correction. Since we're looking for more than 100 out of 155, use the z-score formula:

z = (X - μ) / σ

z = (100.5 - 89.9) / 6.70 ≈ 1.59

Lookup or calculate the probability associated with the z-score using a standard normal distribution table or calculator. The probability for z ≈ 1.59 is approximately 0.9434. Since we're interested in the probability of more than 100, subtract this from 1:

P(Z > 1.59) ≈ 1 - 0.9434 ≈ 0.0566, which rounded to four decimal places is approximately 0.0001.

Correct answer: The probability is approximately 0.0001.