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romanna [79]
3 years ago
10

I’m so confused ahhhhh

Mathematics
2 answers:
barxatty [35]3 years ago
7 0
Hı I m tuncay. I know little englısh. I m turkısh man. Can I help you
Kipish [7]3 years ago
4 0

Answer:

91.................

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Rocky Mountain National Park is a popular park for outdoor recreation activities in Colorado. According to U.S. National Park Se
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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

Step-by-step 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:

\sqrt{V(X)} = \sqrt{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 = \frac{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 = \sqrt{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 = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{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 = \frac{X - \mu}{\sigma}

Z = \frac{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 = \frac{X - \mu}{\sigma}

Z = \frac{89.5 - 81.725}{6.6}

Z = 1.18

Z = 1.18 has a pvalue of 0.8810.

X = 79.5

Z = \frac{X - \mu}{\sigma}

Z = \frac{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 = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{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 = \frac{X - \mu}{\sigma}

Z = \frac{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 = \sqrt{V(X)} = \sqrt{np(1-p)} = \sqrt{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 = \frac{X - \mu}{\sigma}

Z = \frac{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

3 0
3 years ago
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