Use the appropriate normal distribution to approximate the resulting binomial distribution. The manager of Madison Finance Company has estimated that, because of a recession year, 5% of its 400 loan accounts will be delinquent. If the manager's estimate is correct, what is the probability that 11 or more of the accounts will be delinquent?
A) The probability is 0.9891.
B) The probability is 0.9994.
C) The probability is 0.8137.
D) The probability is 0.9854.

Question

Quizplus · Accepted Answer

To approximate the probability using the normal distribution, first calculate the mean ($\mu$) and standard deviation ($\sigma$) of the binomial distribution. With $n = 400$ and $p = 0.05$, $\mu = np = 400 \times 0.05 = 20$ and $\sigma = \sqrt{np(1-p)} = \sqrt{400 \times 0.05 \times 0.95} \approx 4.37$. To find the probability of 11 or more accounts being delinquent, convert 11 to a z-score: $z = \frac{11 - 20}{4.37} \approx -2.06$. Using the z-table, the probability of z being less than -2.06 is approximately 0.0196. Therefore, the probability of 11 or more accounts being delinquent is $1 - 0.0196 = 0.9804$ or 0.9854 when rounded as provided in the options.

Use the Appropriate Normal Distribution to Approximate the Resulting Binomial