Question

 

New York City is the most expensive city in the United States for lodging. The mean hotel room rate is $204 per night. Assume that room rates are normally distributed with a standard deviation of $55.

(a)

What is the probability that a hotel room costs $265 or more per night? (Round your answer to four decimal places.)

(b)

What is the probability that a hotel room costs less than $120 per night? (Round your answer to four decimal places.)

(c)

What is the probability that a hotel room costs between $210 and $300 per night? (Round your answer to four decimal places.)

(d)

What is the cost in dollars of the 10% most expensive hotel rooms in New York City? (Round your answer to the nearest cent.)

Answer 1

a)

µ =    204                          
σ =    55                          
right tailed                              
X ≥   265                          
                              
Z =   (X - µ ) / σ =   1.11                      
                              
P(X ≥   265   ) = P(Z ≥   1.11   ) =   P ( Z <   -1.11   ) =    0.1337(answer)
excel formula for probability from z score is =NORMSDIST(Z)  

b)

µ =    204              
σ =    55              
left tailed                  
X ≤    120              
                  
Z =   (X - µ ) / σ =   -1.53          
                  
P(X ≤   120   ) = P(Z ≤   -1.53   ) =   0.0633(answer)
excel formula for probability from z score is =NORMSDIST(Z)     

c)

µ =    204                              
σ =    55                              
we need to calculate probability for ,                                  
210   ≤ X ≤    300                          
X1 =    210   ,   X2 =   300                  
                                  
Z1 =   (X1 - µ ) / σ =   0.109                          
Z2 =   (X2 - µ ) / σ =   1.745                          
                                  
P (   210   < X <    300   ) =    P (    0.109090909   < Z <    1.745   )
                                  
= P ( Z <    1.745   ) - P ( Z <   0.109   ) =    0.9595   -    0.5434   =    0.4161(answer)
excel formula for probability from z score is =NORMSDIST(Z)

d)

µ =    204                  
σ =    55                  
top 10% = bottom 0.90                  
                      
Z value at    0.9   =   1.282   (excel formula =NORMSINV(α))      
z=(x-µ)/σ                      
so, X=zσ+µ=   1.282   *   55   +   204  
X   =   274.49

so, cost of the 10% most expensive hotel rooms in New York City =$274.49