Scenario A hotel has decided that to cut costs it will outso

Scenario: A hotel has decided that to cut costs, it will outsource its reservation business to another company. The other company claims that it can convert 65% of all incoming calls into reservations for the hotel. The compensation for the outsourcing company is based upon their claimed conversion rate. In an effort to determine if they are getting their money\'s worth from the outsourcing company, the hotel decides to take some samples of calls to see what percentage of reservations is made. Find the 95% confidence Interval for the proportion based upon the following samples. 1. 210 calls with 124 reservations 2. 400 calls with 240 reservations 3. x= 781 and n=1280 4. x = 2460 and n=4000 Question: Do you feel that the outsourcing company is fulfilling their claim based upon these results? Briefly explain your conclusion using 50-100 words.

Solution

a)
CI = p ± Z a/2 Sqrt(p*(1-p)/n)))
x = Mean
n = Sample Size
a = 1 - (Confidence Level/100)
Za/2 = Z-table value
CI = Confidence Interval
Mean(x)=124
Sample Size(n)=210
Sample proportion = x/n =0.5905
Confidence Interval = [ 0.5905 ±Z a/2 ( Sqrt ( 0.5905*0.4095) /210)]
= [ 0.5905 - 1.96* Sqrt(0.0012) , 0.5905 + 1.96* Sqrt(0.0012) ]
= [ 0.524,0.657]

b)
Mean(x)=240
Sample Size(n)=400
Sample proportion = x/n =0.6
Confidence Interval = [ 0.6 ±Z a/2 ( Sqrt ( 0.6*0.4) /400)]
= [ 0.6 - 1.96* Sqrt(0.0006) , 0.6 + 1.96* Sqrt(0.0006) ]
= [ 0.552,0.648]

c)
Mean(x)=781
Sample Size(n)=1280
Sample proportion = x/n =0.6102
Confidence Interval = [ 0.6102 ±Z a/2 ( Sqrt ( 0.6102*0.3898) /1280)]
= [ 0.6102 - 1.96* Sqrt(0.0002) , 0.6102 + 1.96* Sqrt(0.0002) ]
= [ 0.5835,0.6369]

d)
Mean(x)=2460
Sample Size(n)=4000
Sample proportion = x/n =0.615
Confidence Interval = [ 0.615 ±Z a/2 ( Sqrt ( 0.615*0.385) /4000)]
= [ 0.615 - 1.96* Sqrt(0.0001) , 0.615 + 1.96* Sqrt(0.0001) ]
= [ 0.5999,0.6301]