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? (b)What is the probability that at least 60 but less than 70 visitors had a recorded entry through the Beaver Meadows park entrance? (c)What is the probability that fewer than 11 visitors had a recorded entry through the Grand Lake park entrance? (d)What is the probability that more than 50 visitors have no recorded point of entry?

Question
Asked Nov 3, 2019

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?
 
(b)What is the probability that at least 60 but less than 70 visitors had a recorded entry through the Beaver Meadows park entrance?
 
(c)What is the probability that fewer than 11 visitors had a recorded entry through the Grand Lake park entrance?
 
(d)What is the probability that more than 50 visitors have no recorded point of entry?
 
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Expert Answer

Step 1

According to the provided data, the sample size, n is 175. The probability of entering through the Beaver Meadows park entrance is 0.467, the probability of entering through Fall River park entrance is 0.243, the probability of entering through Grand Lake park entrance is 0.063 and the probability of entering through no recorded point is 0.227.

The mean and standard deviation for the binomial to normal approximation of entering through the Beaver Meadows park entrance can be obtained as:

Mean, np
- 175 x 0.467
=81.725
Standard deviation, o np(1-p)
=/175x0.467 x (1-0.467)
= 6.599
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Mean, np - 175 x 0.467 =81.725 Standard deviation, o np(1-p) =/175x0.467 x (1-0.467) = 6.599

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Step 2

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

X-u 65-0.5 -
Using Continuity
P(X2 65) P
correction
X- 64.5-u
- 1- P
64.5-81.725
1-P Z<
6.599
1-P(Z<-2.61)
(Using standard normal table)
- 1-0.0045
=0.9955
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Image Transcriptionclose

X-u 65-0.5 - Using Continuity P(X2 65) P correction X- 64.5-u - 1- P 64.5-81.725 1-P Z< 6.599 1-P(Z<-2.61) (Using standard normal table) - 1-0.0045 =0.9955

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Step 3

(b) The probability that at least 60 but less than 70 visitors had a...

Using Continuity
60-0.5 — м X-и 70-0.5— м
P(60 X< 70) P
correction
59.5-
= P
X- 69.5-
69.5-81.725
59.5-81.725
= P
6.599
6.599
=P(Z <-1.85)-P(Z <-3.36)
(Using standard normal table)
0.0322-0.0004
0.0318
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Image Transcriptionclose

Using Continuity 60-0.5 — м X-и 70-0.5— м P(60 X< 70) P correction 59.5- = P X- 69.5- 69.5-81.725 59.5-81.725 = P 6.599 6.599 =P(Z <-1.85)-P(Z <-3.36) (Using standard normal table) 0.0322-0.0004 0.0318

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