Statistics: Sampling

1. Consider a small population of 4 MBA students who are in the job market. One of the students has no job offers. 1 of the students has 1 job offer and the remaining 2 students each have two job offers.
(a) Find the sampling distribution of the average number of job offers if the sample of size 2 is taken with replacement.
(b) In sampling with replacement, how many different samples of size 5 are there? What is the probability that the average number of job offers in such sample taken at random is equal to (i) zero, (ii) two? (Hint: Don't write all possible samples; instead think about the method you would use to solve the problem.

2. If a customer purchases a book from a display of $3 books at a certain bookstore, the probability that just 1 book is purchased from the display is 1/3, the probability that 2 books are purchases from the display is ½ and the probability that 3 books are purchased from the display is 1/6. If 3 customers are chosen at random (from a very large set) who are independently purchasing books from this particular display, construct the theoretical sampling distribution of the sample mean for the amount of the sale. (Hint: Again, don't write all possible samples; instead think about the methods you'd use to solve the problem.)

3. The number of gallons of milk sold in any given week at a particular grocery store has a normal distribution with mean 4,000 and variance 90,000. Furthermore, this distribution is the same at all stores operated by a particular chain, and the milk sales at any one store are independent of the milk sales at any of the other members of the chain. The chain operates in Fairfield, Westport, Darien and Greenwich with 9 stores in each city.

(a) For a given week, find the probability that a particular store sells at least 4,100 gallon of milk.
(b) For a given week, find the probability that the average number of sales per store in Fairfield is at least 4,100 gallon of milk.
(c) For a given four-week period, find the probability that the average number of sales per store per week for the entire chain of stores is at least 4,100 gallons of milk.

4. The number of candy bars sold from a vending machine on any given day is uniformly distributed with a mean 258 and standard deviation 60, and sales on one day are independent of sale on other days. Consider a period of 144 days.
(a) What is the probability that the sample mean number of candy bars sold for the 144 day period is greater than 263?
(b) What is the probability that the total number of candy bars sold for the 144 day period is greater than 36,000 but less than 38,000?
(c) If the profit per candy bar is 5 cents, what is the probability that the average profit per day over the 144 day period is at least $13?

5. A specification calls for a steel rod to be 3 meters long. When a shipment of rods is received, 25 rods are randomly selected from the shipment (shipments contain 1000 rods). If the average lent hog the 25 rods in the sample is between 9.25 meters and 3.05 meters, the shipment is accepted: otherwise it is returned to the supplier. The length of rods is normally distributed and the standard deviation of the length is known to be .20 meters. If the actual mean length of rods in a given shipment is 2.90 meters what is the probability that the shipment will be returned to the supplier?

6. Tests have shown that consumers generally tend to agree on the strength and sweetness of various fragrances, but they are far less likely to agree on preferences. Suppose that there is no difference in preference regarding the two fragrances among all consumers i.e. half prefer Obsession to Giorgio and the other half prefer Giorgio over Obsession. Suppose that 58 consumers are randomly chosen and tested regarding their preference between the fragrances Obsession and Giorgio.

(a) What is the sampling distribution of the proportion of tested consumers who prefer Obsession?
(b) What is the probability that fewer than 30% of the tested consumers stat a preference for Obsession?
(c) How large must the sample be for the standard deviation of the sample portion to be .015?

7. The number of phone calls to the 800 number for P&G customer service between 8 am and 5 pm, a nine hour day, averages 180 per day, Monday through Friday. For the past several years, approximately 40 % of the calls each day pertained to the moon and stars logo used by P&G and whether or not P&G was devoted to satanic worship. The customer service supervisor is interested in whether those rumors have finally abated. Over a 4-week period she takes a sample of all calls coming in 15 minute periods six different times during the day:

(a) What is the probability that 1/3 or fewer of those fifteen-minute periods had more than calls? (an expert in probability theory calculated that the probability of receiving seven or more calls in any given fifteen-minute interval is.133 and this is independent of what happens before or after the given fifteen-minute interval.
(b) Over the time period studied, the supervisor intercepted 642 calls, 160 of which referred to satanic rituals. Do you think the rumors are abating?