The length of time that a person must wait for an Uber driver to show up is unif

Question

The length of time that a person must wait for an Uber driver to show up is uniformly distributed between 0 and 10 minutes. 60 Uber reservations from a recent weekend were randomly chosen and the waiting times were examined. a. What is the probability that a single individual has to wait more than 5.5 minutes? b. What is the probability that a single individual has to wait more than 8 minutes given that they have already waited for 5 minutes? c. What is the probability that the 60 people in the sample had a mean waiting time of more than 5.5 minutes? d. Are your answers from (a) and (c) the same or different? Why?

Explanation / Answer

a)

for uniform distribution paramter a =0 and b=10

hence probability that a single individual has to wait more than 5.5 minutes =P(X>5.5)=1-P(X<5.5)=1-(5.5-0)/(10-0)

=1-0.55 =0.45

b) probability that a single individual has to wait more than 8 minutes given that they have already waited for 5 minutes =P(X>8|X>5) =P(X>8)/P(X>5) =((10-8)/(10-0))/((10-5)/(10-0)) =2/5 =0.4

c) here sample size is greater than 30 tehrefore sample mean will follow normal distribution

for which mean =(a+b)/2 =(0+10)/2 =5

and std error of mean =((b-a)/(12n))1/2 =(10-0)/(12*60)1/2 =0.3727

for normal distribution z score =(X-)/

d) No they are dififerent as from law of large numbers as sample size increases probability of sample mean to be found near to population mean is more than  farthar away due to reduction in stadnard error of mean.

for normal distribution z score =(X-)/

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