On 100 different days, a traffic engineer counts the number of cars that pass through a certain intersection between 5 P.M. and 5:05 P.M. The results are presented in the following table.

Question

a. Let X be the number of cars passing through the intersection between 5 P.M. and 5:05 P.M. on a randomly chosen day. Someone suggests that for any positive integer x, the probability mass function of X is =(0.2)(0.8)x. Using this function, compute P(X=x) for values of x from 0 through 5 inclusive.

b. Someone else suggests that for any positive integer x, the probability mass function is =(0.4)(0.6)x. Using this function, compute P(X=x)for values of x from 0 through 5 inclusive.

c. Compare the results of parts (a) and (b) to the data in the table. Which probability mass function appears to be the better model? Explain.

d. Someone says that neither of the functions is a good model since neither one agrees with the data exactly. Is this right? Explain.

Answer 1

(a)





(b)





(c) appears to be the better model. Its probabilities are all fairly close to the proportions of days observed in the data. In contrast, the probabilities of 0 and 1 for are much smaller than the observed proportions.



(d) No, this is not right. The data are a simple random sample, and the model represents the population. Simple random samples generally do not reflect the population exactly.