1. (2pt) A variable that takes on the values of 0 or 1 and is used to incorporate the effect of categorical variables in a regression model is called

2. (1pt) The model y= β0+β1x1+β2x12+ε is called

3. (2pt) Which of the following tests is used to determine whether an additional set of variables (more than one independent variables) makes a significant contribution to a multiple regression?

4. (1pt) When dealing with the problem of non-constant variance, the log transformation means using

5. (1pt) A variable such as Z, whose value is Z = X₁X₂ is added to a general linear model in order to account for potential effects of two variables X₁ and X₂ acting together. This type of effect is

6. (1pt) In multiple regression analysis, the word linear in the term "general linear model" refers to the fact that

     7. (1pt) A term used to describe the case when the independent variables in a multiple regression model are correlated is

             a. regression

             b. correlation

             c. multicollinearity

             d. None of the above answers is correct

     8. (1pt) All the independent variables in a multiple regression analysis

             a. must be quantitative

             b. must be either quantitative or categorical but not a mix of both

             c. must assume only positive values

             d. can be either quantitative or categorical

COMP1805AB (Fall 2019)  "Discrete Structures I" Specification for Assignment 1 of 4

Please ensure that you include your name and student number on your submission. Your submission must be created using Microsoft Word, Google Docs, or LaTeX.

1. Translate the following English expressions into logical statements. You must explicitly state what the atomic propositions are (e.g., "Let p be proposition ...") and then show their logical relation.

   1. If it is red then it is not blue and it is not green.

   2. It is white but it is also red and green and blue.

   3. It is black if and only if it is not red, green, or blue.

2. Determine if the following expressions are tautologies, contradictions, or contingencies by using only truth tables. Show all your work and do not skip any steps (i.e., ensure you that you include a new column for every single operation and that you state at the end whether each expression is a tautology, contingency, or contradiction).

   a. ¬(?∧(¬?∨(?→?)))
   b. (? ∨(? ↔(¬?∧ (? ∨?))))

   c. (? ∧¬(? ∧(? ∨(? ↔?))))d. ¬(?∨¬(?→(?∧?)))

3. Determine if the following expressions are tautologies, contradictions, or contingencies by using only the logical equivalences. Show all your work and do not skip any steps (i.e., ensure you that you include the name of each equivalence used (excluding commutativity and associativity) and that you state at the end whether each expression is a tautology, contingency, or contradiction).

   a. (? ∧(? →¬?))b. (¬?→(?∨?))c. (? ↔(?∨ ¬?))d. (¬? ∨(?∧?))

4. Using only the  and the  operators, find a logical expression that is equivalent to (? → (? ∧ p)) → (? ∨ p). For this question, you do not need to specify "how" you found the equivalent expression because you will show both techniques in questions 5 and 6 below.

5. Prove that the expression you found for question 4 above is equivalent to the expression (? → (? ∧ p)) → (? ∨ p) by using only truth tables. Show all your work and do not skip any steps (i.e., ensure that you include a new column for every single operation).

6. Prove that the expression you found for question 4 above is equivalent to the expression (? → (? ∧ p)) → (? ∨ p) by using only the logical equivalences. Show all your work and do not skip any steps.

7. Let L(x) be the predicate "x is a lion", G(x) be the predicate "x is a giraffe", and M(x) be the predicate "x eats meat". Translate the following expressions into English. The universe of discourse is all animals.

   1. ∀? (?(?) ∧ ?(?))

   2. ∃? (¬(?(?) ∧ ?(?)) ∨ ¬?(?))

   3. ∀? ((?(?) ∨ ?(?)) → (¬?(?) → ?(?)))

Scenario/Summary

A confidence interval is a defined range of values such that there is a specified probability that the value of a parameter lies within the interval.

In Part I of this project, you will pick a topic, complete research and provide a write-up that includes calculations. Round all values to two decimalplaces when appropriate.

Deliverables

1. Choose a Topic where you can gather at least 50 pieces of data.

Examples of Topics

1. The Golden Gate Warriors Points Per Game in 2016 (use the points scored in the first 50 games).
2. High School Graduation Rates by State (use the graduation rates for all 50 states)
3. Average Tuition Rates in the US (You have to find the tuition rates of 50 college/universities).
4. The prices of a hotel room per night in a major city (You have to find the price of the same night of hotels in one city).
5. Weights of 50 babies at birth.

2. Write at least a 1-Page Report

Open a Word Document

1. Introduction--Provide a description of your topic and cite where you found your data.
2. Sample Data—Include a 5x10 table including your 50 values in your report. You must provide ALL of your sample data.
3. Problem Computations—For the topic you chose, you must answer the following:

• Determine the mean and standard deviation of your sample.
• Find the 80%, 95%, and 99% confidence intervals.
• Make sure to list the margin of error for the 80%, 95%, and 99% confidence interval.
• Create your own confidence interval (you cannot use 80%, 95%, and 99%) and make sure to show your work. Make sure to list the margin of error.

4. Problem Analysis—Write a half-page reflection.

• What trend do you see takes place to the confidence interval as the confidence level rises? Explain mathematically why that takes place.
• Provide a sentence for each confidence interval created in part c) which explains what the confidence interval means in context of topic of your project.
• Explain how Part I of the project has helped you understand confidence intervals better?
• How did this project help you understand statistics better?

Read the following scenario and complete each of the problems below:

A flashlight company claims that the new bulb in its heavy duty flashlight will average 246 hours of light. A statistics student decides that he/she wants to test this claim at a 5% level of significance to determine if there is evidence to support the claim. The student randomly selects and test 15 flashlight bulbs and records how long the bulb last until it burns out. Assume the life of the bulb is normally distributed. The data is in the table below:

Trial:   1      2             3             4            5     6              7              8             9            10            11           12   13            14           15

Hrs: 246/         224/         231/       242/        234/         240/         243 /        236/         239/        255/        256/ 239/         247/         231/         253

A. Based on a histogram or box-plot of the data, what is the shape of the distribution (normal, skewed left, skewed right, uniform, etc.)? Explain why you can conduct a hypothesis test on the given data. (Compose your answer)

B. Evaluate the claim based on the following assumption: The standard deviation of the population is 7.4 hours. 1. Identify the type of test you will use to test the claim that the flashlight will average 246 hours of light. Explain your answer 2. State the null and alternate hypotheses. 3. Conduct the hypothesis test and determine the p-value. 4. State your conclusion about the claim that the flashlight will average 246 hours of light. 5. Construct a 95% confidence interval for this study.

1.

2.

3.

4.

5.

C. Evaluate the claim based on the following assumption. The population’s standard deviation is not known.

1. Identify the type of test you will use to test the claim that the flashlight will average 246 hours of light. Explain your answer.

2. State the null and alternate hypotheses.

3. Conduct the hypothesis test and determine the p-value.

4. State your conclusion about the claim that the flashlight will average 246 hours of light.

5. construct a 95% confidence interval for this study.

1.

2.

3.

4.

5.

D. Write a 200 to 400 word statement comparing and contrasting the two hypothesis test procedures and results.

Scenario: Upon successful completion of the MBA program, imagine you work in the analytics department for a consulting company. Your assignment is to analyze one of the following databases:

• Manufacturing
• Hospital
• Consumer Food
• Financial

Select one of the databases based on the information in the Signature Assignment Options.

Provide a 1,600-word detailed, four part, statistical report with the following sections:

• Part 1 - Preliminary Analysis
• Part 2 - Examination of Descriptive Statistics
• Part 3 - Examination of Inferential Statistics
• Part 4 - Conclusion/Recommendations

Part 1 - Preliminary Analysis

Generally, as a statistics consultant, you will be given a problem and data. At times, you may have to gather additional data. For this assignment, assume all the data is already gathered for you.

State the objective:

• What are the questions you are trying to address?

Describe the population in the study clearly and in sufficient detail:

• What is the sample?

Discuss the types of data and variables:

• Are the data quantitative or qualitative?
• What are levels of measurement for the data?

Part 2 - Descriptive Statistics

Examine the given data.

Present the descriptive statistics (mean, median, mode, range, standard deviation, variance, CV, and five-number summary).

Identify any outliers in the data.

Present any graphs or charts you think are appropriate for the data.

Note: Ideally, we want to assess the conditions of normality too. However, for the purpose of this exercise, assume data is drawn from normal populations.

Part 3 - Inferential Statistics

Use the Part 3: Inferential Statistics document.

• Create (formulate) hypotheses
• Run formal hypothesis tests
• Make decisions. Your decisions should be stated in non-technical terms.

Hint: A final conclusion saying "reject the null hypothesis" by itself without explanation is basically worthless to those who hired you. Similarly, stating the conclusion is false or rejected is not sufficient.

Part 4 - Conclusion and Recommendations

Include the following:

• What are your conclusions?
• What do you infer from the statistical analysis?
• State the interpretations in non-technical terms. What information might lead to a different conclusion?
• Are there any variables missing?
• What additional information would be valuable to help draw a more certain conclusion?

Assignment #7: One-sample Chi-Square
Directions: Use the Chi-Square option in the Nonparametric Tests menu to answer the questions based on the following scenario. (Assume a level of significance of .05 and use information from the scenario to determine the expected frequencies for each category)

During the analysis of the district data, it was determined that one high school had substantially higher Graduate Exit Exam scores than the state average and the averages of high schools in the surrounding districts. To better understand possible reasons for this difference, the superintendent conducted several analyses. One analysis examined the population of students who completed the exam. Specifically, the superintendent wanted to know if the distribution of special education, regular education, and gifted/talented test takers from the local high school differed from the statewide distribution. The obtained data are provided below.

*For purposes of testing, special education includes any student who received accommodations during the exam.
1. If the student distribution for the local high school did not differ from the state, what would be the expected percentage of students in each category?
2. What were the actual percentages of local high school students in each category? (Report final answer to two decimal places)
3. State an appropriate null hypothesis for this analysis.
4. What is the value of the chi-square statistic?
5. What are the reported degrees of freedom?
6. What is the reported level of significance?
7. Based on the results of the one-sample chi-square test, was the population of test taking students at the local high school statistically significantly different from the statewide population?
8. Present the results as they might appear in an article. This must include a table and narrative statement that reports and interprets the results of the analysis.
Note: The table must be created using your word processing program. Tables that are copied and pasted from SPSS are not acceptable.

Arthur: “George, come into my office for a few minutes. You know that the company brass want to increase the amount of automation in some of our factories. I just got word that this plant will be the first to be automated.”

George: “But the cost and accounting analyses we sent to headquarters last fall showed that it wouldn’t be profitable to make changes like that in this plant. Why did they pick this one?” Arthur: “Apparently, top management wants to try robots and all the high-tech gadgets at one factory, to see if they increase product quality and pay for themselves. They think that in the long run, stockholders will benefit from automation. Anyway, the decision has been made, and it’s our job to make it work. We’re going to have to sell the work force and the community on the decision.”

George: “That won’t be easy. Hundreds of people are going to lose their jobs. There isn’t much else that they can do around here, either.”

Arthur: “Some of the factory people will be able to stay on, if they get some additional training. We can convince the workers and the people in town that the decision was necessary, if we can show them accounting and cost information to justify the decision. If they see good, sound reasoning for the action, they’ll be less likely to resist and cause trouble. We need for them to maintain productivity and efficiency until the new equipment is here. I want you to work on a cost summary we can release to the employees and the town newspaper, showing why automation is a good idea.”

George: “But the net present value and other analyses I did earlier showed this plant should stay the way it is.”

Arthur. “When we were working on the analyses, you said yourself that the benefits of automation are hard to identify and assign numbers to. You had to make several assumptions in order to do those analyses. If you change some of your assumptions, you can make the numbers look better. Try a longer useful life for the new equipment, or change some of the projected cost information. As soon as you have the new numbers, bring them to me to look at.”

Do you agree with the decision of case participants and their opinions? Comment it and make your own recommendations

Read the article below then answer the question that follows.

PHILADELPHIA (WTXF) - The controversial Philadelphia soda tax has started. Sunday, the city began imposing a 1.5 cent per ounce tax on the supply of sweetened beverages to retail dealers, which is supposed to go towards quality pre-K, Community Schools, and better parks, libraries and rec centers.

Among the items taxed: regular and diet sodas, and also teas. The tax is also levied on syrups and concentrates used to make sweetened drinks. And drinks considered “zero calorie” are also taxed.

The tax will be paid by distributors of sweetened beverages, but certainly passed on to customers.

According to the city, a distributor is any person who sells sweetened beverages to a dealer, and a dealer is any person who sells sweetened beverages at retail. Dealers include delis, restaurants, and grocery stores.

For them, the tax will be due the 20th of each month for the prior month. That means the first payment will be due February 20, 2017, for distribution activity in January 2017.

The 1.5-cent-per-ounce tax amounts to 18 cents on a 12-ounce can of soda or $1.44 on a six-pack of 16-ounce bottles. Two cities have soda or soft drink taxes but more than 30 cities and states have rejected them.

On Dec. 19, Common Pleas Court Judge Gary S. Glazer rejected the lawsuit challenging it filed by the American Beverage Association and others.

Lawyers for the beverage industry tried to block the tax, arguing it duplicates the state sales tax already imposed on soda and unfairly taxes drinks based on their size, not price.

But Shanin Specter, of Kline & Specter, PC, representing the plaintiffs in the beverage industry's lawsuit against the city, issued this three-word statement: "We shall appeal."

Suppose that someone has said: “Because the tax is paid by distributors, they will be 18c worse off for each can of soda they sell. That is why they are appealing the tax. Nobody else will be negatively impacted, because after all the distributers are the only ones who pay the tax.”

Explain what is wrong with this person’s statement.

Tax correctly depicted on Demand and Supply model, Diagram/s explained and used to support explanation, Discussion of which parties are impacted by the tax

1. Consider an economy with no international trade, no government spending and no taxes, whose consumption function and investment function are given by the following equations: C = 100,000 + .92Y I = 40,000

a. What is the equilibrium level of aggregate output for this economy?

b. What is the saving function for this economy?

c. Check the solution, as we did in class, by showing that at the equilibrium level of Y total spending exactly matches the level of output. At the equilibrium level of Y, what is the level of saving in the economy?

d. Sketch the solution in a graph. [BTW: PowerPoint has a very useful drawing program that will help you get good-looking diagrams; it is a good thing to know how to use. Go here for a basic tutorial: https://www.youtube.com/watch?v=M5X_zvh2j9Y. After constructing the diagram in PowerPoint you can cut & paste it into a Microsoft Word document.]

e. What is the Keynesian multiplier for this economy (see textbook, chapter 3)? If I increases by $2,000, what will be the new equilibrium level of Y?

2. Now let’s add Government Spending into the model. Government spending is a component of Aggregate Demand, just as Consumption Spending and Investment Spending. For now, let’s not worry about taxes; we can assume the government borrows to pay for all of its spending.

Consider a closed economy with no taxes, whose consumption function, investment level & government spending level are given by the following equations:

C = 5,000 + .80Y I = 9,000 G = 2000 where G represents government spending. The equilibrium condition is, as always, that the value of the economy’s output (Y) must be matched by aggregate demand, but now aggregate demand contains a third element, G. So the equilibrium condition from which we’ll start is now: Y = C + I + G.

a. What is the equilibrium level of aggregate output for this economy?

b. What is the saving function for this economy?

c. Check the solution, as we did in class, by showing that at the equilibrium level of Y total spending exactly matches the level of output. At the equilibrium level of Y, what is the level of saving in the economy?

d. Sketch the solution in a graph.

e. What is the Keynesian multiplier for this economy? If G decreases by $100, what will be the new equilibrium level of Y?