How to find probability using only mean. Do I assume a std. dev. or certain probability distribution here?

Question

The average time a customer spends waiting in the check-out line at Wal-Mart is seven minutes. If wait times are independent and constant, what is the probability the customer will spend between eight and ten minutes waiting in the line?

It DOES NOT give the standard deviation. Am I supposed to assume a certain standard deviation because the term "independent and constant" connotes a certain probability distribution function? With the standard deviation, we could find the z-score and find p that way. Otherwise, is there a different way to solve this problem?

Answer 1

Blaine,

 

You are right that only one parameter is available.  If you assume that check out times occur at a constant average rate then they obey a Poisson process.  The time between check out then follows an exponential distribution as Heather 

stated.  Both these distributions have only one parameter.

 

The exponential distribution:  f= λe^(-λt)  whose cumulative distribution is F=1- e^(-λt)

The parameter, λ, represents the rate in the Poisson process and E(t)=STDEV(t)=1/ λ  as Heather stated.

 

From the question, we determine  λ=1/7

 

We require F(10)-F(8)=e^-8/7-e^-10/7=0.079 (3 sig fig) or about 8 percent

 

Does this make any sense?  We know that 50 percent of the time it takes less than 7 minutes to check out.  If the distribution were uniform, then the chances of checking out in any 1 minute interval would be about 7 percent (7*7=49)...so the answer looks reasonable.