Probability Distribution: Normal, Poisson

1) Trading Volume on the NYSE is heaviest during the first half hour (early morning) and last half hour (late afternoon) of the trading day. The early morning trading volumes (millions of shares) for 13 days in January and February are shown here:
The probability distribution of trading volume is approximately normal.

Trading Volume (millions of shares)
214
202
174
163
198
171
265
212
211
194
201
211
180

a. Compute the mean and standard deviation to use as estimates of the population mean and standard deviation
b. What is the probability that, on a randomly selected day, the early morning trading volume will be less than 180 million shares?
c. What is the probability that, on a randomly selected day, the early morning trading volume will exceed 230 million shares?
d. How many shares would have to be traded for the early morning trading volume on a particular day to be among the busiest 5% of days?

2) The PaperStock Company runs a manufacturing facility that produces a paper product. The fiber content of this product is supposed to be 20 pounds per 1000 square feet. (typical of the paper used in grocery bags, for example.) Because of random variations in the inputs to the process, the fiber content of a roll varies according to a Normal distribution. The variability in fiber content, as measured by standard deviation is 0.1. A given roll of this product MUST be rejected if its actual fiber content is less than 19.8 pounds or greater than 20.3 pounds.

Calculate the probability that a given roll is rejected when the mean is 20 pounds per square feet. Set up a data table showing rejection probability as a function of change in the mean and standard deviation.

3) Suppose the weight of a typical American Male follows a normal distribution with
Mean =M = 180 pounds
Standard deviation= s= 30 pounds

Also, suppose that 91.92% of all American males weigh more than I weigh.

(a) What fraction of American males weigh more than 225 pounds?
(b) How much do I weigh?

4) The weekly demand for Ford car sales follows a normal distribution with mean = 50,000 cars and standard deviation of = 14,000 cars.

(a) There is a 1% chance that Ford will sell more than what number of cars during the next year?

(b) What is the probability that Ford will sell between 2.4 and 2.7 million cars during the next year?

(hint #1:) to find the mean of the annual sales multiply the mean of weekly sales times the number of weeks in a year.
(hint #2:) to find the standard deviation of annual sales multiply the standard deviation of monthly sales by the square root of the number of weeks in a year

5) How many warranty returns?
Because your firm's quality is so high, you expect only 1.3 of your products to be returned, on average, each day for warranty repairs. What are the chances that no products will be returned tomorrow? That one will be returned? How about two? How about three?

a. Find the probability that a Poisson random variable with mean 1.3 is equal to 0,1,2, or 3:
b. Find the probability that a Poisson random variable with mean 1.3 is equal to or less than 2:

6) Customer Arrivals

Suppose customers arrive independently at a constant mean rate of 40 per hour.
(or 1 every 1.5 minutes; = 60/40 = 1.5 customers per minute).
What is the probability that at least one customer arrives in the next five minutes?

(i.e. this is the probability that the exponential waiting time until the next customer
arrives is less than five minutes).

See attached file for full problem description.