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

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... Continue reading "Statistical Concepts: Sampling and Probability" »

Adjusting Events

The following events require adjustments in financial statements:

• Settlement of a court case
• Asset impairments (asset not well accounted for)
• Determination of costs of assets purchased or sold before the reporting period
• Determination of profit or bonus payment
• Fraud or errors

Non-Adjusting Events

The following events do not require adjustments in financial statements:

• Major business combinations or disposal of a subsidiary (sell of a subsidiary)
• Plan to discontinue operations
• Purchases of assets or expropriation by government
• Destruction of a plant (by a fire, earthquake…)
• Restructuring
• Ordinary shares transactions
• Changes in asset prices or foreign exchange rates after reporting period
• Changes in tax rates or tax law
• New commitments or

5 Meanings of Fractions

Part-Whole

It goes well beyond shading a region. The circle model is particularly effective in illustrating the part-whole relationship. Focus: How many parts.

Measurement

It involves identifying a length and then using that length as a measurement piece to determine the length of an object. Focus: How much rather than how many parts.

Division

This is often not connected to fractions, which is unfortunate. Consider the idea of sharing $10 with 4 people; each person will receive 1/4 of the money or $2.50.

Operator

Fractions can be used to indicate an operation, as in 4/5 of 20 squares. This situation indicates a fraction of a whole number, and students might be able to use mental math to solve it. Knowing how to represent fractions... Continue reading "Understanding Fractions: Meanings, Models, and Applications" »

Fundamental Logarithm Properties

These are key cancellation properties of logarithms and exponentials:

• e^ln8 = 8
• ln(e⁴) = 4
• log₈(8³) = 3
• 5^log5(2) = 2

Essential Logarithm Rules

• Product Rule: log_a(xy) = log_a(x) + log_a(y)
• Quotient Rule: log_a(x/y) = log_a(x) - log_a(y)
• Power Rule: log_a(x^p) = p log_a(x)
• Change of Base Formula: log_B(D) = log(D) / log(B) (using common log) or ln(D) / ln(B) (using natural log)
• Logarithmic to Exponential Form: log_b(n) = a is equivalent to n = b^a.

Logarithm Definitions and Undefined Cases

• A logarithm answers the question: "What is the power?" For example, log_b(n) asks "To what power must b be raised to get n?"
• Logarithms are undefined for:
  • log(0)
  • log(-#) (logarithm of a negative number)
• The natural logarithm: log_e(x) = ln(x).
• Exponential

Key Terms in Linear Programming & Optimization

Constraint

An equation or inequality that rules out certain combinations of decision variables as feasible solutions.

Problem Formulation

The process of translating a verbal statement of a problem into a mathematical statement called the mathematical model.

Mathematical Model

A representation of a problem where the objective and all constraint conditions are described by mathematical expressions.

Decision Variable

A controllable input for a linear programming model.

Objective Function

The expression that defines the quantity to be maximized or minimized in a linear programming model.

Nonnegativity Constraints

A set of constraints that requires all variables to be nonnegative.

Linear Program

A mathematical... Continue reading "Linear Programming Terminology: Core Concepts Defined" »

Sensitivity Analysis: The study of how changes in the coefficients of a linear programming problem affect the optimal solution.

Objective Function Coefficient Allowable Increase (Decrease): The allowable increase (decrease) of an objective function coefficient is the amount the coefficient may increase (decrease) without causing any change in the values of the decision variables in the optimal solution. The allowable increase/decrease for the objective function coefficients can be used to calculate the range of optimality.

Objective Coefficient Range (Range of Optimality): The range of values over which an objective function coefficient may vary without causing any change in the values of the decision variables in the optimal solution.

Shadow

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

Polygon Interior Angles Theorem

The sum of the measures of interior angles of an n-gon is (n-2)x180

Interior angles of a quadrilateral:

The sum of the measures of the interior angles of a quadrilateral is 360^o

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is 360.

Angles have to measure up to 360. 360/n (n=#of sides)

Theorems

If a quadrilateral is a parallelogram, then its opposite sides are congruent

If a quadrilateral is a parallelogram, then its opposite angles are congruent

If a quadrilateral is a parallelogram, then its consecutive angles are supplementary

If a quadrilateral is a parallelogram, then its diagonal bisects each other

If both pairs of opposite sides of a quadrilateral

Econometric Notation: Residuals and Estimates

Residuals are denoted by e (Sample Regression Function, SRF) and the true error term by u (Population Regression Function, PRF).

• Coefficients: Estimated coefficients (e.g., $\hat{\beta}_0$ and $\hat{\beta}_1$) in the SRF are denoted with hats (caps). True coefficients ($\beta_0, \beta_1$) in the PRF are without hats.
• Dependent Variable: The predicted value $\hat{Y}_t$ (the value derived from the regression equation without the residual) is denoted with a hat. The actual observed value $Y_t$ (which includes the residual or error term) is without a hat.

Fundamentals of Hypothesis Testing

Hypothesis testing makes inferences about the validity of specific economic (or other) theories from a sample of the... Continue reading "Econometric Hypothesis Testing and Regression Model Specification" »

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