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VLD [36.1K]
3 years ago
15

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night (USA Today

, April 30, 2012). Assume that room rates are normally distributed with a standard deviation of $55.a. What is the probability that a hotel room costs $225 or more per night (to 4 decimals)?b. What is the probability that a hotel room costs less than $140 per night (to 4 decimals)?c. What is the probability that a hotel room costs between $200 and $300 per night (to 4 decimals)?d. What is the cost of the 20% most expensive hotel rooms in New York City? Round up to the next dollar.
Mathematics
1 answer:
Ierofanga [76]3 years ago
5 0

Answer:

a) We can find the z score for the value of 225 and we got:

z = \frac{225-204}{55}=0.382

And we can find this probability with the complement rule:

P(z>0.382)=1-P(z

b) z = \frac{140-204}{55}=-1.164

And we can find this probability using the normal standard table or excel and we got:

P(z

c) P(200

And we can find this probability with this difference:

P(-0.072

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

P(-0.072

d) For this part we want to find a value a, such that we satisfy this condition:

P(X>a)=0.2   (a)

P(X   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.8 of the area on the left and 0.2 of the area on the right it's z=0.842. On this case P(Z<0.842)=0.8 and P(z>0.842)=0.2

If we use condition (b) from previous we have this:

P(X  

P(z

But we know which value of z satisfy the previous equation so then we can do this:

z=0.842

And if we solve for a we got

a=204 +0.842*55=250.31

Step-by-step explanation:

Previous concepts

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".  

Part a

Let X the random variable that represent the costs of a population, and for this case we know the distribution for X is given by:

X \sim N(204,55)  

Where \mu=204 and \sigma=55

We are interested on this probability

P(X>225)

And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\sigma}

We can find the z score for the value of 225 and we got:

z = \frac{225-204}{55}=0.382

And we can find this probability with the complement rule:

P(z>0.382)=1-P(z

Part b

P(X

And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\sigma}

If we apply this formula to our probability we got this:

z = \frac{140-204}{55}=-1.164

And we can find this probability using the normal standard table or excel and we got:

P(z

Part c

P(200

And the best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\sigma}

If we apply this formula to our probability we got this:

P(200

And we can find this probability with this difference:

P(-0.072

And in order to find these probabilities we can find tables for the normal standard distribution, excel or a calculator.  

P(-0.072

Part d

For this part we want to find a value a, such that we satisfy this condition:

P(X>a)=0.2   (a)

P(X   (b)

Both conditions are equivalent on this case. We can use the z score again in order to find the value a.  

As we can see on the figure attached the z value that satisfy the condition with 0.8 of the area on the left and 0.2 of the area on the right it's z=0.842. On this case P(Z<0.842)=0.8 and P(z>0.842)=0.2

If we use condition (b) from previous we have this:

P(X  

P(z

But we know which value of z satisfy the previous equation so then we can do this:

z=0.842

And if we solve for a we got

a=204 +0.842*55=250.31

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