Question

enorio national park in costa rica has a roughly consistent year-round climate. on any given day, we assume there is a 40% chance of heavy rain. a. we are interested in forecasting the number of heavy rain days during 2019. write down n and p if we are to use the binomial distribution for this forecast. b. calculate the mean and standard deviation of heavy rain days in 2019 using the binomial model. use r or a calculator. c. find the probability that the park will experience exactly 145 days of heavy rain in 2019. d. find the probability that the park will see between 125 and 175 days of heavy rain in 2019. e. the yearly amount of rainfall in the park is normally distributed with mean 200 inches and standard deviation of 20 inches. find the probability that the park will experience more than 230 inches of rain in 2019.

Answer 1

To use the binomial distribution for forecasting the number of heavy rain days in 2019 at the Enorio National Park in Costa Rica, n represents the number of trials (365) and p represents the probability of heavy rain on any given day (0.4). The mean of heavy rain days in 2019 is 146 days and the standard deviation is approximately 8.49 days. The probability of exactly 145 heavy rain days in 2019 is approximately 0.0099, and the probability of experiencing between 125 and 175 heavy rain days is approximately 0.9996. The probability of experiencing more than 230 inches of rain in 2019 is approximately 0.9332.

Step-by-step explanation:

a. To use the binomial distribution for forecasting the number of heavy rain days in 2019 at the Enorio National Park in Costa Rica, we need to define the values of n and p.

n represents the number of trials or days in this case, which is 365 (as there are 365 days in a year).

p represents the probability of success on a single trial, which is the chance of heavy rain on any given day. Here, p is 0.4 or 40% (as stated in the question).

b. To calculate the mean of heavy rain days in 2019, multiply n (365) by p (0.4) to get 146 days. The standard deviation can be found using the formula: sqrt(n * p * (1 - p)), which gives us a standard deviation of approximately 8.49 days.

c. To find the probability of exactly 145 days of heavy rain in 2019, we can use the binomial probability formula: P(X=k) = nCk * p^k * (1-p)^(n-k), where X is the random variable representing the number of heavy rain days. Plugging in the values, the probability is approximately 0.0099.

d. To find the probability of experiencing between 125 and 175 days of heavy rain in 2019, we need to calculate the cumulative probability from 125 to 175. We can sum up the individual probabilities for each number of days within that range. Alternatively, we can use a binomial calculator or software to find the cumulative probability, which is approximately 0.9996.

e. The annual amount of rainfall in the park is given to be normally distributed with a mean of 200 inches and a standard deviation of 20 inches. To find the probability of experiencing more than 230 inches of rain in 2019, we can calculate the z-score using the given values: z = (x - mean) / standard deviation. Plugging in the values, we get z = (230 - 200) / 20 = 1.5. Using a Z-table or software, we can find the probability corresponding to a z-score of 1.5, which is approximately 0.9332.