Given the following information, formulate an inventory management system. The item is demanded 50 weeks a year.

a. Determine the order quantity and reorder point. (Use Excel’s NORMSINV( ) function to find your z-value and then round that z-value to 2 decimal places. Do not round any other intermediate calculations. Round your final answers to the nearest whole number.)

b. Determine the annual holding and order costs. (Do not round any intermediate calculations. Round your final answers to 2 decimal places.)

c. Assume a price break of $50 per order was offered for purchase quantities of 2,200 units per order. If you took advantage of this price break, how much would you save annually? (Do not round any intermediate calculations (including number of setups per year). Round your final answer to 2 decimal places.)

(10 pts) Suppose that when I drive to school, I encounter one traffic light on Lewis Road and one traffic light on Santa Rosa Rd. Let the random variable X = number of red lights that I encounter on Lewis and Y = number of red lights that I encounter on Santa Rosa. Suppose that the marginal distributions of X and Y are as shown in the following probability table:

X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0
Notice that E(X) = E(Y) = .5, and Var(X) = Var(Y) = .25.

a) Fill in the table in such a way that Corr(X,Y) = 1. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0


b) Fill in the table in such a way that Corr(X,Y) = -1. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0


c) Fill in the table in such a way that Corr(X,Y) = 0. Verify that indeed it checks out.
X=-1 X=1 Total
Y=-1 0.5
Y=1 0.5
Total 0.5 0.5 1.0

Consider the variable W=X+Y, representing the total number of red lights I encounter on my drive to school.
d) Calculate E(W)





e) For each of the cases in parts a), b) and c), calculate SD(W)

The manufacturer of a certain electronic component claims that they are designed to last just slightly more than 4 years because they believe that customers typically replace their device before then. Based on information provided by the company, the components should last a mean of 4.24 years with a standard deviation of 0.45 years. For this scenario, assume the lifespans of this component follow a normal distribution

3) The company considers a component to be “successful” if it lasts longer than the warranty period before failing. They estimate that about 70.3% of components last more than 4 years. They find a random group of 10 components that were sold and count the number of them which were “successful,” lasting more than 4 years.

a. What is the expected number of these 10 components that will last more than 4 years? (Round your answer to 1 decimal place.)

b. What is the standard deviation for the number of these 10 components that will last more than 4 years? (Round your answer to 1 decimal place.)

c. What is the probability that no more than 6 of these components will last more than 4 years?

4) The company is not satisfied with how many returns they are processing, and accountants in the company are recommending that they change the warranty period. The accountants suggest basing the period of the warranty on making sure, in the long run, only about 5% of customers will return the component. What amount of time corresponds to the shortest 5% of lifespans for this component? (Round your answer to 1 decimal place.)

9. At a certain school, 41% of the students play soccer, 30% play volleyball, and 14% play both soccer and volleyball. If a student is chosen at random, find the probability that he/she plays neither soccer nor volleyball.: *
(A) 0.71
(B) 0.57
(C) 0.43
(D) 0.413

Questions 10-12: A simple linear regression model was fit to the situation of using the number of pages in a book (in hundreds) to predict the number of typos in the book. The equation is y = 1.2 + 3.4x.

10. Interpret the slope.: *
(A) For every additional page in length, a book is expected to have an extra 3.4 typos on average.
(B) A 400-page book, on average, should have 3.4 more typos than a 300-page book.
(C) For every additional 3.4 pages in length, a book is expected to have an extra 1.2 typos on average.
(D) The slope has no practical interpretation in this context.

11. Find the predicted number of typos in a 500-page book.: *
(A) 17
(B) 18.2
(C) 171.2
(D) 1701.2

12. Explain what it would mean if an actual 500-page book had a residual of -3.2.: *
(A) This particular book is definitely an outlier and should be dropped from the model.
(B) This particular book had a predicted value smaller than its actual value.
(C) This particular book had a predicted value larger than its actual value.
(D) A mistake has been made since residuals cannot be negative.

It is surprising (but true) that if 23 people are in the same room, there is about a 50% chance that at least two people will have the same birthday. Suppose you want to estimate the probability that if 30 people are in the same room, at least two of them will have the same birthday. You can proceed as follows.

a. Generate random birthdays for 30 different people. Ignoring the possibility of a leap year, each person has a 1/365 chance of having a given birthday (label the days of the year 1 to 365). You can use the RANDBETWEEN function to generate birthdays. What do you expect the average birthday (a number between 1 and 365) among the 30 people be?

b. Once you have generated 30 people's birthdays, how can you tell whether at least two people have the same birthday? One way is to use Excel's RANK function. (You can learn how to use this function in Excel's online help.) This function returns the rank of a number relative to a given group of numbers. In the case of a tie, two numbers are given the same rank. For example, if the set of numbers is 4, 3, 2, 5, the RANK function returns 2, 3, 4, 1. (By default, RANK gives 1 to the largest number.) If the set of numbers is 4, 3, 2, 4, the RANK function returns 1, 3, 4, 1. What do you expect the sum of the birthday ranks for the 30 people be, if there are no two people with the same birthday?

Many freeways have service (or logo) signs that give information on attractions, camping, lodging, food, and gas services prior to off-ramps. These signs typically do not provide information on distances. An article reported that in one investigation, six sites along interstate highways where service signs are posted were selected. For each site, crash data was obtained for a three-year period before distance information was added to the service signs and for a one-year period afterward. The number of crashes per year before and after the sign changes were as follows.

a)The article included the statement "A paired t test was performed to determine whether there was any change in the mean number of crashes before and after the addition of distance information on the signs." Carry out such a test. [Note: The relevant normal probability plot shows a substantial linear pattern.]
State and test the appropriate hypotheses. (Use

α = 0.05.)

Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to three decimal places.)

b)If a seventh site were to be randomly selected among locations bearing service signs, between what values would you predict the difference in number of crashes to lie? (Use a 95% prediction interval. Round your answers to two decimal places.)

1.) A boat capsized and sank in a lake. Based on an assumption of a mean weight of 148 ​lb, the boat was rated to carry 60 passengers​ (so the load limit was 8 comma 880 ​lb). After the boat​ sank, the assumed mean weight for similar boats was changed from 148 lb to 170 lb. Complete parts a and b below.

a.) Assume that a similar boat is loaded with 60 ​passengers, and assume that the weights of people are normally distributed with a mean of 178.3 lb and a standard deviation of 40.9 lb. Find the probability that the boat is overloaded because the 60 passengers have a mean weight greater than 148 lb.

b. The boat was later rated to carry only 15 ​passengers, and the load limit was changed to 2 comma 550 lb. Find the probability that the boat is overloaded because the mean weight of the passengers is greater than 170 ​(so that their total weight is greater than the maximum capacity of 2 comma 550 ​lb). The probability is nothing. ​(Round to four decimal places as​ needed.) Do the new ratings appear to be safe when the boat is loaded with 15 ​passengers? Choose the correct answer below.

A. Because there is a high probability of​ overloading, the new ratings appear to be safe when the boat is loaded with 15 passengers.

B. Because there is a high probability of​ overloading, the new ratings do not appear to be safe when the boat is loaded with 15 passengers.

C. Because the probability of overloading is lower with the new ratings than with the old​ ratings, the new ratings appear to be safe.

D. Because 178.3 is greater than 170​, the new ratings do not appear to be safe when the boat is loaded with 15 passengers.

2.) An engineer is going to redesign an ejection seat for an airplane. The seat was designed for pilots weighing between 120 lb and 171 lb. The new population of pilots has normally distributed weights with a mean of 130 lb and a standard deviation of 29.4 lb.

a. If a pilot is randomly​ selected, find the probability that his weight is between 120 lb and 171 lb.

The probability is approximately ​(Round to four decimal places as​ needed.)

b. If 33 different pilots are randomly​ selected, find the probability that their mean weight is between 120 lb and 171 lb.

The probability is approximately. ​(Round to four decimal places as​ needed.)

c. When redesigning the ejection​ seat, which probability is more​ relevant?

A. Part​ (b) because the seat performance for a single pilot is more important.

B. Part​ (a) because the seat performance for a single pilot is more important.

C. Part​ (a) because the seat performance for a sample of pilots is more important.

D. Part​ (b) because the seat performance for a sample of pilots is more important.

1. There are multiple sections of 3 classes that have to be offered. Class A has an average enrollment of 120 per term. Class B has an average of 100 students per term and Class C has an average enrollment of 85 students per term. The rules for opening and cancelling sections are as follows:

1. Any section of this class must have at least 15 students to run. Any less than this and the section is cancelled.
2. Each section has a maximum enrollment of 45 students.
3. A new section will open up when the previous section (s) have an average enrollment of 40.
4. A new section will open up to accommodate department regulations on staffing.

For each class run a simulation using your chosen distribution and determine the following:

1. Given that anytime enrollment in a class reaches over 175 students, there will be 5 sections of class, what is the probability of this happening with each class?

Class A =

Class B =

Class C =

ii)    Given that anytime enrollment in a class reaches 135 students to 180 students, there will be 4 sections of class, what is the probability of this happening with each class?

Class A =

Class B =

Class C =

3. Given that anytime enrollment in a class reaches 95 students to 120 students, there will be 3 sections of class, what is the probability of this happening with each class?

Class A =

Class B =

Class C =

4. If a third section of Class C opens when enrollment goes above 80, and it gets cancelled when enrollment is below 95, How often does the third section get cancelled? (Enrollment is between 80 and 95)

5. Based upon the probabilities you found above, how many sections of each class do you offer? 2, 3, 4 or 5?

Class A =

Class B =

Class C =

Using the number of sections, you found above create a schedule that maximizes the quality of teaching these classes. Full Time profs must teach 3-4 sections. Part time profs must teach 1-2 sections.

6. What is the minimum number of sections (total) that need to be offered?

7. What is the maximum number of sections (total) you can offer?

Professor Data is below:

Given your answers in part v, and possibly modified by your answers in part vi, what is the quality score of your department’s teaching?

How many sections of each class do the professors teach?

1.         The minimum probability of rejecting the null hypothesis that is most acceptable in the field of science is .05. Why are researchers generally less willing to raise the probability of rejecting the null hypothesis to levels that are greater than .05?

A television factory knows that 2% of their televisions are defective. In a batch of 100 new televisions, 4 of them are inspected at random. What is the probability that 1 of them will be defective? What is the probability that they all turn out to be defective?