A popular resort hotel has 300 rooms and is usually fully booked. About 4% of the time a reservation...

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A popular resort hotel has 300 rooms and is usually fully booked. About 4% of the time a reservation is canceled before 6:00 p.m. deadline with no penalty. What is the probability that at least 280 rooms will be occupied? Use binomial distribution to find the exact value and the normal approximation to the binomial and compare your answers. 2. The number and frequency of Atlantic hurricanes annually from 1940 through 2007 is shown here. NUMBER XXXXXXXXXXFrequency XXXXXXXXXX1 a) Find the probabilities of 0-8 hurricanes each season using data. b) Assuming a Poisson distribution and using the mean number of hurricanes per season from the empirical data, compute the probabilities of experiencing 0-8 hurricanes in a season. Compare these to your answer to part (a). How good does a Poisson distribution model this phenomenon? 3. The distribution of SAT scores in math for an incoming class of business students has a mean of 580 and standard deviation of 25. Assume that the scores are normally distributed. a) Find the probability that an individual’s score is less than 550. b) Find the probability that an individual’s score is between 560 and 600. c) Find the probability that an individual’s score is greater than 620. d) What % of students will have scored better than 700? e) Find the standardized values for students scoring 500, 600, 700 on the test. 4. Historical data show that customers who download music from a popular web service spend approximately $20 per month, with a standard deviation of $4. Find the probability that a customer will spend at least $15 per month. If the company samples 100 customers, find the mean and standard deviation of the number who spend at least $15 per
	
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QUANTITATIVE METHODS MSC603 
Probability Distributions and Data Modeling 
 
1. A popular resort hotel has 300 rooms and is usually fully booked. About 4% of the time a 
reservation is canceled before 6:00 p.m. deadline with no penalty. What is the probability that 
at least 280 rooms will be occupied? Use binomial distribution to find the exact value and the 
normal approximation to the binomial and compare your answers. 
 
2. The number and frequency of Atlantic hurricanes annually from 1940 through 2007 is shown 
here. 
NUMBER XXXXXXXXXX 
Frequency XXXXXXXXXX1 
a) Find the probabilities of 0-8 hurricanes each season using data. 
b) Assuming a Poisson distribution and using the mean number of hurricanes per season from 
the empirical data, compute the probabilities of experiencing 0-8 hurricanes in a season. 
Compare these to your answer to part (a). How good does a Poisson distribution model this 
phenomenon? 
 
3. The distribution of SAT scores in math for an incoming class of business students has a mean of 
580 and standard deviation of 25. Assume that the scores are normally distributed. 
a) Find the probability that an individual’s score is less than 550. 
b) Find the probability that an individual’s score is between 560 and 600. 
c) Find the probability that an individual’s score is greater than 620.  
d) What % of students will have scored better than 700? 
e) Find the standardized values for students scoring 500, 600, 700 on the test. 
 
4. Historical data show that customers who download music from a popular web service spend 
approximately $20 per month, with a standard deviation of $4. Find the probability that a 
customer will spend at least $15 per month. If the company samples 100 customers, find the 
mean and standard deviation of the number who spend at least $15 per month. What is the 
probability that at least 40% of them will spend a t least $15 per month?

Robert · Accepted Answer

1. A popular resort hotel has 300 rooms and is usually fully booked. About 4% of the time a 
reservation is canceled before 6:00 p.m. deadline with no penalty. What is the probability that 
at least 280 rooms will be occupied? Use binomial distribution to find the exact value and the 
normal approximation to the binomial and compare your answers. 
Solution: Probability that the rooms are occupied = 0.96 
      Probability that the reservation is cancelled = 0.04 
      Using the binomial distribution as binomdist with n = 300, p = 0.96, q = 0.04) 
  (   )  ∑(
 
 
)  (   )
Prob (x ≥ 280) = ∑ (   
   
)(    )   (    )
     
 = 0.9899 
Probability (At least 280 rooms will be occupied) = 0.9899 
             Now, we use the normal approximation to binomial distribution.., 
Mean = np = 300*0.96 = 288 
SD = sqrt(npq)= sqrt(300*0.96*0.04) = 3.394 
Since n is large, we can use approximation to normal distribution. 
X ≥ 280 
   
    
 
  
       
     
        
P (z ≥ -2.36) = 0.9909 
Therefore, the probability that at least 280 rooms are booked is 0.9909
When we compare the values of the probability by two different methods, we see that 
approximately they are same if we retain the values till 2 decimal places. 
Thus, approximation value is little higher than the exact probability value.
2.

A popular resort hotel has 300 rooms and is usually fully booked. About 4% of the time a reservation is canceled before 6:00 p.m. deadline with no penalty. What is the probability that at least 280...

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