Notes, summaries, assignments, exams, and problems for Mathematics

1. The t-test is never valid if the sample size is small, say less than 30. (TRUE OR FALSE)

It can be valid if the sample is approximately normal.

2. The results of an observational study indicate that people who use vitamin supplements get fewer colds than people who don't. However, we can't conclude that vitamin supplements prevent colds because this type of study does not allow us to infer causation. (TRUE OR FALSE)

Only experimental designs can infer causation because it allows us to create treatment groups that are very similar.

3. A survey was administered to a random sample of college students. Both males and females were surveyed, and one question asked was "How much are you willing to spend on a stereo system (in dollars)?" An analysis

Distribution of Moody's Ratings

Historical Default Rates

• Rating Agencies
• Default rates from bonds
• Market Data

CAPM Model for Bond Returns

The usual CAPM has to be modified to include expected loss per year.

What must the β be to justify the excess spread for Baa Bonds?

Merton's Model

A Structural Model That Predicts Default Rates

How Do We Find V₀ and σ_V?

We know E₀ and σ_E, and

An application of Ito's lemma shows

These allow us to find V₀ and σ_V from E₀ and σ_E.

Default Correlation

Defaults: If Firm A Defaults, is Firm B More Likely to Default Too?

No Default Correlation

Consider two firms, X and Y:

Q_X = 0.1, Q_Y = 0.2

If X and Y are uncorrelated, then the probability that they both default is Q_X × Q_Y = 0.02

Building Default Correlation

If X and Y are correlated,...

Qualitative Variables

Nominal Variables

Nominal variables are qualitative variables that cannot be ordered in an ascending or descending manner; that is, they cannot be ranked. For example, blood group.

Ordinal Variables

Ordinal variables are variables that can be ordered in an ascending or descending manner; that is, they can be ranked.

Quantitative Variables

Discrete Variables

Discrete variables are variables whose values are obtained by counting.

Continuous Variables

Continuous variables are variables whose values are obtained by measurement using a scale.

Mean

Advantages

• Has many good theoretical properties
• Used as the basis of many statistical tests
• Good summary statistic for symmetrical distribution
• Easy to calculate
• Possible for further algebraic treatment

Disadvantages

• Less

Checks: A Comprehensive Overview

What is a Check?

A check is a document instructing a bank (the drawee) to pay a specific amount of money to the check holder (the payee) from the account of the person who wrote the check (the drawer).

Parties Involved

• Drawer: The person who writes and signs the check, authorizing the payment.
• Drawee: The bank or financial institution where the drawer has an account.
• Payee/Holder: The person or entity to whom the check is made payable.
• Guarantor: A person who guarantees payment if the drawer's account has insufficient funds.
• Endorser: The payee who signs the back of the check to transfer ownership.
• Endorsee: The person to whom the check is endorsed.

Types of Checks

Forms of Writing

• Bearer: Payable to the person holding

Sampling Stage

Sample Size:

• Depends on population size
• The type of sample
• Feature (parameter) of the population that is analyzed
• The maximum permissible error in the estimation of the parameters

The Form: The information collected can be classified as:

• Acts or behaviors that can be compared
• Information: analyzes the degree of respondents' knowledge on specific topics
• Opinions or Judgments: such as perceived service quality
• Attitudes or predispositions of mind: We are looking for what is thought in relation to something
• Motives or explanations for specific behaviors: the question is, why certain views or acts?
• Possible future behavior: it may ask whether or not to consume a prepared product.

Concept and Structure of Questionnaires

The questionnaire is a way...

) Hypotheses, Type I and Type II Errors, and Statistical Tests

For each of the following tests, state the hypotheses, identify if it is a right-, left-, or two-tailed test and write Type I Error and Type II Error pertaining to the problem.

1. a) A professor of statistics states that the average student spends 3 hours studying for the midterm exam. Ho: μ = 3 vs. Ha: μ ≠ 3 Two-tailed test Type I Error: Concluding the average student doesn’t study 3 hours for midterm when in fact that is false. Type II Error: Not concluding the average student doesn’t study 3 hours for midterm when in fact that is true.
2. b) A spouse stated that the average amount of money spent on Christmas gifts for immediate family members is above $1,200. Ho: μ = 1200 (or

This document outlines the steps to calculate production costs, including equivalent units and costs for work-in-progress (WIP) and finished goods.

Grammar and Vocabulary

1. I have lost my pen. Can I borrow yours?
2. As soon as I have more information, I will call you.
3. Did you know she was pregnant?
4. We have had this can since last summer.
5. When I lived in Hungary, I had to take a bus to go to work.
6. Would you mind answering my questions?
7. 80% of our electronic components are produced in the Far East.
8. The manager and his staff usually meet once a week.
9. What are you responsible for?
10. When did you arrive at the hostel?
11. The way of managing companies in this country is very different from ours.
12. Has she answered your email yet?
13. You will have to work hard if you really want to pass the exam.
14. He never drinks any alcohol when he goes to parties.
15. Have you got something to declare?
16. The plane was four hours late.
17. Driving too

6) Distribution channels for automobiles in Japan and the United States are similar in that most car buyers in both countries visit dealerships before making a purchase.

8) Kodak adopted a direct involvement approach in Japan by utilizing independent agents, distributors and retailers.

9) A distributor with good contacts is obviously the best one to select in terms of generating quick sales and revenues.

10) When utilizing the services of a local independent distributor, management is well advised to treat the distributor as a temporary market-entry vehicle.

11) An exporter that is new to a particular market would do well to choose a middleman with a reputation for "cherry picking" products.

12) When Carrefour, Tesco, or Walmart set up shop in developing...

Parameter vs. Statistic

Parameter: Population (μ, σ)

Statistic: Sample (x̄, s)

Rules of Probability

1. 0 ≤ P(x) ≤ 1
2. Σ P(x) = 1
3. P(not x) = 1 - P(x)

Central Limit Theorem

As n (sample size) gets bigger, the sample distance will become approximately normal (shape).

Law of Large Numbers

As n gets bigger, the sample mean (x̄) will get closer to the population mean (μ) (number).

68-95-99.7 Rule

• 68% of data will lie within 1 standard deviation of the mean (σ + μ)
• 95% of data will lie within 2 standard deviations of the mean (2σ + μ)
• 99.7% of data will lie within 3 standard deviations of the mean (3σ + μ)

Sampling Types

Simple Random Sampling (SRS): Normal, random picking.

Systematic Sampling: Every kth sample.

Stratified Sampling: Groups are put together...