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 8 visitors had a recorded entry through the Grand Lake park entrance?

(d)

What is the probability that more than 40 visitors have no recorded point of entry?

Answer 1

n=

175

p=

0.4670

a)

here mean of distribution=μ=np=		81.73
and standard deviation σ=sqrt(np(1-p))=		6.60
for normal distribution z score =(X-μ)/σx

probability =P(X>84.5)=P(Z>(84.5-81.725)/6.6)=P(Z>0.42)=1-P(Z<0.42)=1-0.6628=0.3372

b)

probability =P(79.5<X<89.5)=P((79.5-81.725)/6.6)<Z<(89.5-81.725)/6.6)=P(-0.34<Z<1.18)=0.881-0.3669=0.5141

c)

probability =P(X<7.5)=(Z<(7.5-11.025)/3.214)=P(Z<-1.1)=0.1357

d)

probability =P(X>40.5)=P(Z>(40.5-39.725)/5.541)=P(Z>0.14)=1-P(Z<0.14)=1-0.5557=0.4443