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 85 visitors had a recorded entry through the Beaver Meadows park entrance?(b) What is the probability that at least 80 but less than 90 visitors had a recorded entry through the Beaver Meadows park entrance?(c) What is the probability that fewer than 12 visitors had a recorded entry through the Grand Lake park entrance?(d) What is the probability that more than 55 visitors have no recorded point of entry?

Answer 1

a) 0.6628 = 66.28% probability that at least 85 visitors had a recorded entry through the Beaver Meadows park entrance

b) 0.5141 = 51.41% probability that at least 80 but less than 90 visitors had a recorded entry through the Beaver Meadows park entrance

c) 0.5596 = 55.96% probability that fewer than 12 visitors had a recorded entry through the Grand Lake park entrance.

d) 0.9978 = 99.78% probability that more than 55 visitors have no recorded point of entry

Explanation:

Using the normal approximation to the binomial.

Binomial probability distribution

Probability of exactly x sucesses on n repeated trials, with p probability.

Can be approximated to a normal distribution, using the expected value and the standard deviation.

The expected value of the binomial distribution is:

`E(X) = np`

The standard deviation of the binomial distribution is:

`√(V(X)) = √(np(1-p))`

Normal probability distribution

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

When we are approximating a binomial distribution to a normal one, we have that
`\mu = E(X)`,
`\sigma = √(V(X))`.

In this problem, we have that:

175 visitors, so
`n = 175`

a)

46.7% through the Beaver Meadows, so
`p = 0.467`

`\mu = E(X) = np = 175*0.467 = 81.725`

`\sigma = √(V(X)) = √(np(1-p)) = √(175*0.467*0.533) = 6.6`

This probability, using continuity correction, is
`P(X \geq 85 - 0.5) = P(X \geq 84.5)`, which is 1 subtracted by the pvalue of Z when X = 84.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (84.5 - 81.725)/(6.6)`

`Z = 0.42`

`Z = 0.42` has a pvalue of 0.6628.

66.28% probability that at least 85 visitors had a recorded entry through the Beaver Meadows park entrance.

b)

This is
`P(80 - 0.5 \leq X < 90 - 0.5) = P(79.5 \leq X \leq 89.5)`, which is the pvalue of Z when X = 89.5 subtracted by the pvalue of Z when X = 79.5. So

X = 89.5

`Z = (X - \mu)/(\sigma)`

`Z = (89.5 - 81.725)/(6.6)`

`Z = 1.18`

`Z = 1.18` has a pvalue of 0.8810.

X = 79.5

`Z = (X - \mu)/(\sigma)`

`Z = (79.5 - 81.725)/(6.6)`

`Z = -0.34`

`Z = -0.34` has a pvalue of 0.3669.

0.8810 - 0.3669 = 0.5141

0.5141 = 51.41% probability that at least 80 but less than 90 visitors had a recorded entry through the Beaver Meadows park entrance

c)

6.3% over the Grand Lake park entrance, so
`p = 0.063`

`\mu = E(X) = np = 175*0.063 = 11.025`

`\sigma = √(V(X)) = √(np(1-p)) = √(175*0.063*0.937) = 3.2141`

This probability is P(X < 12 - 0.5) = P(X < 11.5), which is the pvalue of Z when X = 11.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (11.5 - 11.025)/(3.2141)`

`Z = 0.15`

`Z = 0.15` has a pvalue of 0.5596.

55.96% probability that fewer than 12 visitors had a recorded entry through the Grand Lake park entrance.

d)

22.7% with no recorded point, so
`p = 0.227`

`\mu = E(X) = np = 175*0.227 = 39.725`

`\sigma = √(V(X)) = √(np(1-p)) = √(175*0.227*0.773) = 5.54`

This probability is
`P(X \leq 55 + 0.5) = P(X \leq 55.5)`, which is the pvalue of Z when X = 55.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (55.5 - 39.725)/(5.54)`

`Z = 2.85`

`Z = 2.85` has a pvalue of 0.9978

99.78% probability that more than 55 visitors have no recorded point of entry