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?

The mortality experience of 8146 male employees of a research, engineering, and metal-fabrication plant in Tonawa- nda, New York, was studied from 1946 to 1981 [2]. Potential workplace exposures included welding fumes, cutting oils, asbestos, organic solvents, and environmental ionizing ra- diation, as a result of waste disposal during the Manhattan Project of World War II. Comparisons were made for specific causes of death between mortality rates in workers and U.S. white-male mortality rates from 1950 to 1978.

Suppose that 17 deaths from cirrhosis of the liver were observed among workers who were hired prior to 1946 and who had worked in the plant for 10 or more years, whereas 6.3 were expected based on U.S. white-male mortality rates.

7.48 What is the SMR for this group?

7.49 Perform a significance test to assess whether there is an association between long duration of employment and mortality from cirrhosis of the liver in the group hired prior to 1946. Report a p-value.

To test whether extracurricular activity is a good predictor of college success, a college administrator records whether students participated in extracurricular activities during high school and their subsequent college freshman GPA.

(a) Code the dichotomous variable and then compute a point-biserial correlation coefficient. (Round your answer to three decimal places.)

2. A psychologist noted that people have more difficulty sleeping in a bright room than in a dark room. She measured whether the intensity of the light could predict the time it took a sample of 4 participants to fall asleep. The data for this hypothetical study are listed in the following table.

Compute an analysis of regression for this hypothetical study. (Round your answers to two decimal places.)

3.

opy the table to Excel then for each part using Excel formula to do the questions below and make sure to apply DCOVA, which stands for Define, Collect, Organize, Visualize, and Analyze.

1. Find the mean and standard deviation,

2. What did you observe and what that means in term of Cholesterol levels for a patient.

3. Find the five-number summary ( Minimum, Q1, Q2, Q3, and Maximum)

4. Interpret your result in term of cholesterol level

5. Find interquartile range (IQR) and what that means?

6. Draw a box-and-whiskers plot for data given

7. Do you have any outliers? Explain the method used to identify the outliers

The shape of the distribution of the time required to get an oil change at a 15​-minute oil-change facility is unknown.​ However, records indicate that the mean time is 16.4 minutes and the standard deviation is 3.5 minutes. To compute probabilities regarding the sample mean using the normal​ model, what size sample would be​ required?

The table shows the relationship between the production and the export of rice in vietnam from 1985 to 2000.

How much rice would you expect Vietnam to export in 2015 if the production that year is 4225000 tonnes?

How can use a scatter plot to find the linear model

How can you use your model to make a prediction?