Use the appropriate normal distribution to approximate the resulting binomial distribution.
A marksman's chance of hitting a target with each of his shots is 60%. If he fires 30 shots, what is the probability of his hitting the target between 12 and 21 times, inclusive?

A) The probability is 0.9064.
B) The probability is 0.8954.
C) The probability is 0.8920.
D) The probability is 0.4647.

Question

Use the appropriate normal distribution to approximate the resulting binomial distribution. ​
A marksman's chance of hitting a target with each of his shots is 60%. If he fires 30 shots, what is the probability of his hitting the target between 12 and 21 times, inclusive?
​
A) The probability is 0.9064.
B) The probability is 0.8954.
C) The probability is 0.8920.
D) The probability is 0.4647.

Quizplus · Accepted Answer

Since the marksman has a 60% chance of hitting the target with each shot, we can use a normal approximation to the binomial distribution with mean np = 30(0.6) = 18 and standard deviation sqrt(np(1-p)) = sqrt(30(0.6)(0.4)) = 2.1909.

To find the probability of hitting the target between 12 and 21 times, we can standardize these values using the formula z = (x - np) / sqrt(np(1-p)), where x is the number of hits we're interested in. Then we can use the standard normal distribution table or calculator to find the probability.

For x = 12: z = (12 - 18) / 2.1909 = -2.735
For x = 21: z = (21 - 18) / 2.1909 = 1.372

Using the standard normal distribution table, we find that the probability of z being between -2.735 and 1.372 is 0.8954. Therefore, the answer is B.

Use the Appropriate Normal Distribution to Approximate the Resulting Binomial