Question

Delivery Time	Percent customers looking for newspaper by time	Percent undelivered at time
6:00 AM	15%	38%
6:15 AM	5%	21%
6:30 AM	8%	22%
6:45 AM	6%	14%
7:00 AM	15%	7%
7:15 AM	12%	6%
7:30 AM	16%	4%
7:45 AM	3%	3%
8:00 AM	5%	3%
"Anytime"	15%	0%

The Herald marketing department is seeking to increase home-delivery sales through an aggressive direct-marketing campaign that includes mailings, discount coupons, and telephone solicitations. Feedback from these efforts indicates that getting their newspapers delivered early in the morning is a very important factor for both prospective and existing subscribers. After several brainstorming sessions, a team consisting of members from the marketing and circulation departments decided that guaranteeing newspaper delivery by a specific time could be an important selling point in retaining and getting new subscribers. The team concluded that the Herald should offer a guarantee that customers will receive their newspaper by a certain time or else that day’s issue is free. To assist the team in setting a guaranteed delivery time, Al Leslie, the research director, determined that the circulation department had data that showed the percentage of newspapers yet undelivered every quarter hour from 6AM to 8AM. Jan Shapiro remembered that customers were asked on their subscription forms at what time they would be looking for their copy of the newspaper to be delivered. These data were subsequently combined and are available below. Your job is to review the internal data and propose a reasonable time (to the nearest quarter hour) to guarantee delivery. To help explore the effects of your choice, calculate the following probability: If a sample of 50 customers is selected on a given day, what is the probability, given your selected delivery time, that a. Fewer than 3 customers will receive a free newspaper? b. 2, 3, or 4 customers will receive a free newspaper? c. More than 5 customers will receive a few newspaper? Use the binomial distribution function in excel to perform required calculations.

Answer 1

Let selected delivery time =6:00 AM

Sample of 50 customers selected on a given day.

Among them, number of customers looking for newspaper by time =50*15% =7.5 which is rounded up to 8

So, n =8

Success =undelivered at time =receiving free newspaper

Proportion of success (receiving freenews paper), p =38% =0.38

Distribution: Binomial

Formula: P(X =r) =C(n, r).p^r.(1-p)^n-r

(where P= Probability; r =number of successes).

Excel Formula:

For probability mass function, P(X =r):

=BINOMDIST(number of successes, number of independent trials, probability of success, FALSE)

For cumulative distribution function, P(X r):

=BINOMDIST(number of successes, number of independent trials, probability of SUCCESS, TRUE)

a.

P(Fewer than 3 customers) =P(X < 3) =P(X 2) =0.3585 =35.85%

b.

P(X =2, 3 or 4) =P(X =2)+P(X =3)+P(X =4) =0.7268 =72.68%

c.

P(more than 5) =P(X > 5) =1 - P(X 5) =1 - 0.9615 =0.0385 =3.85%