1. A salesperson makes four calls per day. A sample of 100 days gives the following frequencies of sales volumes.

Records show sales are made to 30% of all sales calls. Assuming independent sales calls, the number of sales per day should follow a binomial distribution. Assume that the population has a binomial distribution with n = 4, p =.25, and x = 0, 1, 2, 3, and 4.

1. Compute the expected frequencies for x = 0, 1, 2, 3, and 4 by using the binomial probability function. Combine categories if necessary to satisfy the requirement that the expected frequency is five or more for all categories.
2. Use the goodness of fit test to determine whether the assumption of a binomial distribution should be rejected. State the Hypotheses and the conclusion. Use α = .10. Note: Because no parameters of the binomial distribution were estimated from the sample data, the degrees of freedom are k-1 where k is the number of categories.

PLEASE NO HANDWRITTEN ANSWERS, AND PLEASE WITH EXCEL FORMULAS!

Accidental Death & Dismemberment Insurance covers accidents that results in death or loss of limbs due to accidents. Considering dismemberments only, let a discrete random variable X be the number of limbs lost from a policy holder in a given year.

Suppose that the probability distribution of X is x

0 p(x)=0.80

1 p(x)=0.13

2 p(x)=0.04

3 p(x)=0.02

4 p(x)=0.01

(a) Find the expected number of limbs lost for a randomly selected policy holder in a given year.

(b) suppose the insurance company pays $10,000 for loss of one limb, $20,000 for loss of two limbs, $50,000 for loss of three limbs, and $100,000 for loss of all limbs, how much should the annual premium be if the insurance company wants an average of $50 profit per policy holder? Hint: What is the expected insurance payment for an arbitrary policy holder?

Isle Royale, an island in Lake Superior, has provided an important study site of wolves and their prey. Of special interest is the study of the number of moose killed by wolves. In the period from 1958 to 1974, there were 296 moose deaths identified as wolf kills. The age distribution of the kills is as follows.

(a) For each age group, compute the probability that a moose in that age group is killed by a wolf. (Round your answers to three decimal places.)

(b) Consider all ages in a class equal to the class midpoint. Find the expected age of a moose killed by a wolf and the standard deviation of the ages. (Round your answers to two decimal places.)

Benford's Law claims that numbers chosen from very large data files tend to have "1" as the first nonzero digit disproportionately often. In fact, research has shown that if you randomly draw a number from a very large data file, the probability of getting a number with "1" as the leading digit is about 0.301. Suppose you are an auditor for a very large corporation. The revenue report involves millions of numbers in a large computer file. Let us say you took a random sample of n =351 numerical entries from the file and r = 108 of the entries had a first nonzero digit of 1. Let p represent the population proportion of all numbers in the corporate file that have a first nonzero digit of 1. Test the claim that p is less than 0.301 by using α = 0.05. What is the level of significance?

Select one: a. 0.025 b. 0.05 c. 0.975 d. 0.95 e. 0.1

There is a box with space for 16 items. Let A denote the number of things that
are type one and B the number of things that are
type two. Assume that A and B are independent
random variables. Assume that all possible (a,b)
pairs are equally likely.
I) How many possible pairs (a,b) are there?
II) Which event is more likely {A = 1} or {B = 0}?
Justify your answer.
III) Compute P(B=5) and P(A=10)
IV) If there are 10 things type 1, what is the
probability that there are 6 things type 2
in the box?
V) Compute P(B=5 | A=10)
VI) Compute P(B <= 5) and P(A <= 10)
VII) Compute P(B=5 | A <= 10)
VIII) Compute P(A+B=5) and P(A+B = 10)
IX) Compute P(A+B <= 5) and P(A+B <= 10)
X) Compute P(A+B <= 5 | A+B <= 10)
XI) Compute E(A+B)

Mr. Beautiful, an organization that sells weight training sets, has an ordering cost of $40 for the BB-1 set (BB-1 stands for Body Beautiful Number 1). The carrying cost for BB-1 is $5 per set per year. To meet demand, Mr. Beautiful orders large quantities of BB-1 4 times a year. The stockout cost for BB-1 is estimated to be $10 per set. Over the past several years, Mr. Beautiful has observed the following demand during the lead time for BB-1:

The reorder point for BB-1 is 60 sets. What level of safety stock should be maintained for BB-1?

The optimal quantity of safety stock which minimizes expected total cost is ____sets (enter your response as a whole number).

Isle Royale, an island in Lake Superior, has provided an important study site of wolves and their prey. Of special interest is the study of the number of moose killed by wolves. In the period from 1958 to 1974, there were 296 moose deaths identified as wolf kills. The age distribution of the kills is as follows.

(a) For each age group, compute the probability that a moose in that age group is killed by a wolf. (Round your answers to three decimal places.)

(b) Consider all ages in a class equal to the class midpoint. Find the expected age of a moose killed by a wolf and the standard deviation of the ages. (Round your answers to two decimal places.)

For the population of Cal Poly students, let X = number of hours slept in the last 24 hours and Y = number of exams today. Consider the following (admittedly simplistic) joint probability distribution for Xand Y.

For your convenience, I have calculated the following: ?(?)=6.24,?(?2)=40.34,?(?)=0.89,?(?2)=1.55E(X)=6.24,E(X2)=40.34,E(Y)=0.89,E(Y2)=1.55.

A.    Determine the marginal distribution of X.

B.    Calculate E(XY).

C.    Calculate the correlation between X and Y. Don’t round at intermediate steps. Give your final answer to three decimal places.

D.    Interpret the correlation value in the context of this example.

E.    Are X and Y independent random variables? How can you tell?

Your boss was so happy to learn that you just completed a lecture on the applications of probability distributions in decision making last night. She called you to her office and asked you to help her decide which of the two comparable supercomputers A or B she should purchase.

Supercomputer A costs $14,000 and supercomputer B costs $14,800. Your company replaces supercomputers every three years. The repair contract for supercomputer A costs $70 per month and covers and unlimited number of repairs. The repair contract for supercomputer B costs $200 per repair. Based on past performance, the distribution of the number of repairs over any one-year period for supercomputer B is shown below:

Your boss asked that you give a recommendation based on the overall cost as to which supercomputer A or B along with its repair contract should be purchased. What would your recommendation be? Give a statistical justification to support your recommendation.