Problem 4-03 (Algorithmic)

The employee credit union at State University is planning the allocation of funds for the coming year. The credit union makes four types of loans to its members. In addition, the credit union invests in risk-free securities to stabilize income. The various revenue producing investments together with annual rates of return are as follows:

The credit union will have $2.4 million available for investment during the coming year. State laws and credit union policies impose the following restrictions on the composition of the loans and investments.

• Risk-free securities may not exceed 30% of the total funds available for investment.
• Signature loans may not exceed 10% of the funds invested in all loans (automobile, furniture, other secured, and signature loans).
• Furniture loans plus other secured loans may not exceed the automobile loans.
• Other secured loans plus signature loans may not exceed the funds invested in risk-free securities.

How should the $2.4 million be allocated to each of the loan/investment alternatives to maximize total annual return? Round your answers to the nearest dollar.

What is the projected total annual return? Round your answer to the nearest dollar.

$ ______________________

You are getting your wisdom teeth removed and are a participant in a clinicaltrial for three different procedures, one of which is randomly assigned to you. Of those whoundergo the first procedure, 8% get an infection; of those who undergo the second procedure,4% get an infection; of those who undergo the third procedure, 9% get an infection.

(a) What is the probability that you will not get an infection.

b) Unfortunately, you got an infection after removing your wisdom teeth! What is the probability that you were assigned to the first procedure?

The owner of a popular chicken restaurant, Chicken-For-Me, with many branches wanted to know if the quality of customer service at a new restaurant was acceptable. One aspect of service that was examined was the length of time that customers had to wait in line before ordering their food. The restaurant decided on acceptable probabilities for the waiting-time categories, and these are given below.
Waiting-time Category
Probability
No more than 1 minute
0.15
More than 1 minute but no more than 3 mins
0.30
More than 3 minutes but no more than 5 mins
0.24
More than 5 minutes but no more than 10 minutes
0.25
More than 10 minutes
0.06
To investigate whether the quality of customer service was acceptable, waiting times were recorded for a random sample of 100 customers at the new Chicken-for-Me. The table below shows the number of customers observed in the five waiting-time categories.
Waiting-time Category
  
Number of Customers
No more than 1 minute
20
More than 1 minute but no more than 3 mins
31
More than 3 minutes but no more than 5 mins
31
More than 5 minutes but no more than 10 minutes
15
More than 10 minutes
3
Total
100
Use the sample data for the 100 customers to conduct a statistical test to determine if the waiting times at the new Chicken-For-Me are inconsistent with the acceptable probabilities for the waiting- time categories.
Question #2
A randomly selected group of men and women were surveyed to investigate the association between gender and the amount of money spent at a local store, Bullseye. Results are shown in the table below:
Dollars Spent @ “Bullseye”
$0 to $50
$51 to $100
$101 to $200
more than $201
Total
Men
18
85
71
90
264
Women
35
72
98
142
347
Total
53
157
169
232
611
Is there convincing evidence that there is an association between gender and the amount of money spent at “Bullseye”?

Problem 7-9 Dixie Showtime Movie Theaters, Inc., owns and operates a chain of cinemas in several markets in the southern U.S. The owners would like to estimate weekly gross revenue as a function of advertising expenditures.

Data for a sample of eight markets for a recent week follow. Market Weekly Gross Revenue ($100s) Television Advertising ($100s) Newspaper Advertising ($100s) Mobile 102.5 5.1 1.6 Shreveport 52.7 3.2 3 Jackson 75.8 4 1.5 Birmingham 127.8 4.3 4 Little Rock 137.8 3.5 4.3 Biloxi 101.4 3.6 2.3 New Orleans 237.8 5 8.4 Baton Rouge 219.6 6.9 5.8

(a) Use the data to develop an estimated regression with the amount of television advertising as the independent variable. Let x represent the amount of television advertising. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) = + x Test for a significant relationship between television advertising and weekly gross revenue at the 0.05 level of significance. What is the interpretation of this relationship? The input in the box below will not be graded, but may be reviewed and considered by your instructor.

(b) How much of the variation in the sample values of weekly gross revenue does the model in part (a) explain? If required, round your answer to two decimal places. %

(c) Use the data to develop an estimated regression equation with both television advertising and newspaper advertising as the independent variables. Let x1 represent the amount of television advertising. Let x2 represent the amount of newspaper advertising. If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) = + x1 + x2 Test whether each of the regression parameters β0, β1, and β2 is equal to zero at a 0.05 level of significance. What are the correct interpretations of the estimated regression parameters? Are these interpretations reasonable?

(d) How much of the variation in the sample values of weekly gross revenue does the model in part (c) explain? If required, round your answer to two decimal places. % (e) Given the results in part (a) and part (c), what should your next step be? Explain. (f) What are the managerial implications of these results?

suppose that you encounter two traffic lights on your commute to school. Based on past experience, you judge that the probability is .60 that the first light will be red when you get to it, .50 that the second light will be red, and .40 that both lights will be red.

a)Determine the conditional probability that the second light will be red, given that the first light is red. (Here and throughout, show the details of your calculations.)

b)Are the events {first light is red} and {second light is red} independent? Justify your answer.

c) Given that at least one light is red, what is the probability that both lights are red? (Show your work.)

In a study of texting speed, 45 adults aged 18-22 were randomly chosen. Each person was asked to type the following phrase exactly “Hi! What’s up?” on a cell phone. The average time it took was 2.72 seconds. This was found to happen with about a 25% chance when compared to someone saying it takes less than 3 seconds to type.

1.) Identify the following: a. Population: ______________________________________________

b. Sample: _________________________________________________

c. Unit/individual: ___________________________________________

d. Response variable: ________________________________________

2.) What type of variable is the response variable? And what is the level of measurement?

3.) Determine whether the results below appear to have statistical significance, and also determine whether the results have practical significance.

In a study of a hair growth vitamin, 65 subjects grew an average of 1.4 inches of hair in 4 weeks. If regular hair growth is ¼ inch per month. It is found that there is about a 2% chance of getting such results with a diet that has no effect.

i. Does the weight loss program have statistical significance?

ii. Does the weight loss program have practical significance?

Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate for the data below.

4) Alert system of yellow (lowest), orange, and red (highest) right

5) Social security numbers

6) Years in which a war was started

7) Class times measured in minutes

Problem 1. The purpose of this problem is to practice the use of logic operations and quantifiers. For each Statement X below determine if each of the three statementsX1, X2, X3 that follow it satisfy the following:

a) Xi implies X;
b) X implies Xi;
c) if Xi is true then X must be false; d) if X is true then Xi must be false.

Statement A. In every house there is a mouse.

A1. There is no house without a mouse.A2. There exists a house without a mouse.A3. Mice don’t live in houses.

Statement B. For every mouse there is a blouse, such that if the mouse wears the blouse he’ll get a gift from Carl Friedrich Gauss.

B1. There is a mouse that can wear any blouse, but still won’t get a gift from Gauss.

B2. There are no mice for which there does not exist a special blouse, such that if the mouse is not getting a gift from Gauss it means that he did not wear that blouse.B3. If a mouse did not get a gift from Gauss, it must be that he hasn’t tried on all

the blouses yet.
Statement C. If Statement A is true then Statement B is true.

C1. In every house there is a mouse that never wore a blouse, but got a gift from Gauss.

C2. Every house has at least 3 mice, but under no condition would Gauss give something to a mouse.

C3. There is a mouse in my house that likes to wear a silver blouse and got some cookies from my spouse. (My name’s Johanna Gauss).

The Fox TV network is considering replacing one of its prime-time crime investigation shows with a new family-oriented comedy show. Before a final decision is made, network executives commission a sample of 500 viewers. After viewing the comedy, 250 indicated they would watch the new show and suggested it replace the crime investigation show.

A) Estimate the value of the population proportion. (Round the z-values to 2 decimal places. Round your answer to 3 decimal places.)

B) Develop a 90% confidence interval for the population proportion. (Use z Distribution Table.) (Round the z-values to 2 decimal places. Round your answers to 3 decimal places.)

Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 A.M. to 10:00 P.M. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour, but the temp agency managing the part time staff will charge the steakhouse one extra dollar per hour for shifts starting after 3:00 PM. The total number of full-time and part-time employees needed varies with the time of day as shown.

One full-time employee comes on duty at 11:00 A.M., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 P.M. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern.

1. Develop a minimum-cost schedule for part-time employees. What is the total payroll for the part-time employees? If required, round your answer to the nearest dollar.
   
   Total daily salary cost = $  
   
   How many part-time shifts are needed?
   
   
   
   Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.
   
   The input in the box below will not be graded, but may be reviewed and considered by your instructor.
   
   
   
2. Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. If your answer is zero, enter "0".
   
   Optimal schedule for part-time employees:
   

   Starting Time	Number of part- time employees (4-hour shifts)	Number of part- time employees (3-hour shifts)
   11:00 A.M.
   12:00 P.M.
   1:00 P.M.
   2:00 P.M.
   3:00 P.M.
   4:00 P.M.
   5:00 P.M.
   6:00 P.M.
   7:00 P.M.

   
   How many part-time shifts are needed?
   
   
   
   What is the cost savings compared to the previous schedule? If required, round your answer to the nearest cent.
   
   Total cost reduced to $  .

In a clinical study of a test devised to detect colorectal cancer it was found that 13% of people without cancer received a positive result (false positive) and 8% of people with cancer received a negative result (false negative). According to the American Cancer Society, the lifetime risk of developing colorectal cancer is about 1 in 22 (4:49%) for men and 1 in 24 (4:15%) for women. According to the World Bank the population of the U. S. is 50:52% female. If we simplify the model assuming that there are only two genders, male and female, (a) what is the probability that a person with undisclosed gender will develop colorectal cancer in their lifetime? (b) what is the probability that a person with undisclosed gender has colorectal cancer, given that they took the test twice and the results were positive the first time and negative the second time?

Using these 5 core concepts on reinterpreting correlations, please explain all 5 of them.

reciprocal causation

causal variable

extraneous variable

mediating variable

moderating variable

The weight of a product is normally distributed with a mean 10 ounces. A randomly selected unit of this product weighs 13 ounces. The probability of a unit weighing more than 13 ounces is 0.0014. The production supervisor has lost files containing various pieces of information regarding this process including the standard deviation. Determine the value of standard deviation for this process.

Five people on the basement of a building get on an elevator that stops at seven floors. Assuming that each has an equal probability of going to any floor, find

(a) the probability that they all get off at different floors

(b) the probability that two people get off at the same floor and all others get off at different floors.

Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui . 1. [3 points] What are the assumptions of this model so that the OLS estimators are BLUE (best linear unbiased estimates)? 2. [4 points] Let βˆ 1 and βˆ 2 be the OLS estimators of β1 and β2. Derive βˆ 1 and βˆ 2. 3. [2 points] Show that βˆ 2 is an unbiased estimator of β2.

The​ quality-control manager at a compact fluorescent light bulb​ (CFL) factory needs to determine whether the mean life of a large shipment of CFLs is equal to 7,487 hours. The population standard deviation is 1,080 hours. A random sample of 81 light bulbs indicates a sample mean life of 7,187 hours. a. At the 0.05 level of​ significance, is there evidence that the mean life is different from 7,487 hours? b. Compute the​ p-value and interpret its meaning. c. Construct a 95​% confidence interval estimate of the population mean life of the light bulbs. d. Compare the results of​ (a) and​ (c). What conclusions do you​ reach?