Question

Rocky Mountain National Park is a popular park for outdoor recreation activities in Colorado. According to U.S. National Park Service statistics, 46.7% of visitors to Rocky Mountain National Park in 2018 entered through the Beaver Meadows park entrance, 24.3% of visitors entered through the Fall River park entrance, 6.3% of visitors entered through the Grand Lake park entrance, and 22.7% of visitors had no recorded point of entry to the park.† Consider a random sample of 175 Rocky Mountain National Park visitors. Use the normal approximation of the binomial distribution to answer the following questions. (Round your answers to four decimal places.)

(a)What is the probability that at least 65 visitors had a recorded entry through the Beaver Meadows park entrance?

Answer 1

To find the probability that at least 65 visitors had a recorded entry through the Beaver Meadows park entrance, we can use the normal approximation of the binomial distribution. Calculating the mean and standard deviation, we can then use the z-score formula to find the probability.

Step-by-step explanation:

To find the probability that at least 65 visitors had a recorded entry through the Beaver Meadows park entrance, we can use the normal approximation of the binomial distribution. We can assume that the number of visitors entering through the Beaver Meadows park entrance follows a normal distribution with a mean of np and a standard deviation of sqrt(npq), where n is the sample size, p is the probability of success, and q is the probability of failure. In this case, n = 175 and p = 0.467.

First, we calculate the mean (np) and standard deviation (sqrt(npq)):

• Mean = 175 * 0.467 = 81.725
• Standard Deviation = sqrt(175 * 0.467 * (1 - 0.467)) = 7.703

Next, we can use the z-score formula to calculate the probability:

P(X >= 65) = 1 - P(X < 65) = 1 - P(Z < (65 - 81.725)/7.703)

Using a standard normal distribution table or a calculator, we can find the probability associated with the calculated z-score and subtract it from 1 to find the final probability. The result is the probability that at least 65 visitors had a recorded entry through the Beaver Meadows park entrance.