Cars arrive at Wendy's drive-through at a rate of 0.2 car per minute between the hours of 11:00 p.m. and 1:00 a.m. on Saturday evening. Wendy's begins an advertising blitz that touts its late-night service. After one week of advertising, Wendy's officials count the number of cars, X, arriving at Wendy's drive-through between the hours of 12:00 midnight and 12:30 a.m. at 200 of its restaurants. The results are shown in the following table:
x (number of cars arriving) Frequency
1..................................................................4
2..................................................................5
3................................................................13
4................................................................23
5................................................................25
6................................................................28
7................................................................25
8................................................................27
9................................................................21
10...............................................................15
11.................................................................5
12.................................................................3
13.................................................................2
14.................................................................2
15.................................................................0
16.................................................................2
(a) Construct a probability distribution for the random variable X, assuming it follows a Poisson process with λ = 0.2 and t = 30. This is the probability distribution of X before the advertising.
(b) Compute the expected number of restaurants that will have 0 arrivals, 1 arrival, and so on.
(c) Compare these results with the number of arrivals after the advertising. Does it appear the advertising was effective? Why?