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

Chapter 1: Foundations of Statistics

Data: Information derived from observations, counts, measurements, or responses.

Statistics: The science of collecting, organizing, analyzing, and interpreting data to make informed decisions.

Population: The collection of all outcomes, responses, measurements, or counts of interest.

Sample: A subset or part of a population.

Parameter: A numerical description of a population characteristic.

Statistic: A numerical description of a sample characteristic.

Descriptive Statistics: Methods to organize, display, and summarize data (e.g., mean, range, graphs, tables).

Inferential Statistics: Using sample data to draw conclusions about a population.

Qualitative Data: Attributes, labels, or non-numerical entries.

Quantitative

Introduction: A Data Science Journey

My name is Amit Kadam, and I currently reside in Mumbai. I completed my Bachelor of Engineering (B.E.) degree in 2021. After graduation, the pandemic limited job opportunities, and my family faced financial challenges, so I took my first opportunity at Sterling as a Senior Associate, where I worked for 2.5 years.

Initially, I was responsible for document verification, but I was soon promoted to manage drug health screening processes. In this role, I handled candidate health reports, prepared data for analysis, and developed strong attention to detail and data-handling skills.

During this time, a friend who successfully transitioned into data science encouraged me to explore the field. I started by learning... Continue reading "Data Science Career Transition & Predictive Modeling" »

Question Bank #1 – Net Value Functions

L03 – Engineering Economics & Net Value Applications

Review Questions

Recall the nanoRIMS example discussed in lecture. If the net value of buying the nanoparticles is $0 (the reference), determine the net value per week of having a grad student make the nanoparticles based on the following information:

• Benefit = $896/week
• Cost:
  • Cost of consumable supplies per week: Ingredients & electricity to make one batch as accurately as a grad student does is $5/100 mL * 200 mL/week = $10/week
  • Cost of time: Grad student time is $15/hr * 9 hours/100 mL * 200 mL/week = $270/week
  • Cost of space: Occupying a whole fume hood space for 16 hours during working time is $12.50/hr * 16 hrs/week = $200/week
  • Cost of any device:

What is a Scatter Diagram?

Definition

A scatter diagram (or scatter plot) is a graphical representation of two variables where each point represents an observation consisting of paired values from two datasets. The horizontal axis (X-axis) represents one variable, and the vertical axis (Y-axis) represents the other.

Construction

Each point (x_i, y_i) is plotted on the graph for the corresponding values of the two variables.

Utility in Correlation Analysis

Scatter diagrams are essential for:

• Visualizing relationships: Helps identify if a linear or non-linear relationship exists.
• Direction of correlation:
  • Positive correlation: As X increases, Y increases (points slope upwards).
  • Negative correlation: As X increases, Y decreases (points slope downwards).

14. Obtain the regression equation of Y onX and correlation coefficient for the following: X; 4 6 8 10 12 , f ;7 9 8 12 15 1. Calculate the necessary sums: X Y X² XY 2 10 4 20 3 9 9 27 7 11 49 77 8 8 64 64 10 12 100 120 ΣX = 30 ΣY = 50 ΣX² = 226 ΣXY = 308 Export to Sheets 2. Calculate the slope (b): b = (nΣXY - ΣXΣY) / (nΣX² - (ΣX)²) where n is the number of data points (n = 5 in this case) b = (5 * 308 - 30 * 50) / (5 * 226 - 30²) b = (1540 - 1500) / (1130 - 900) b = 40 / 230 b ≈ 0.174 3. Calculate the y-intercept (a): a = (ΣY - bΣX) / n a = (50 - 0.174 * 30) / 5 a = (50 - 5.22) / 5 a ≈ 8.956 4. The fitted line: Substitute the values of a and b into the equation Y = a + bX: Y = 8.956 + 0.174X Therefore, the fitted straight... Continue reading "Regression Equation and Probability Addition Theorem Solutions" »

--------------------------------   

  ----------------------------------------------------------

Write an interpretation of r^2 using the template in the Activity 2.1 Readings. We will do this one as a class.

Template: The proportion of the variation in the Y variable that is explained by the SLR model with the X variable is r^2.

For slope : Template: As x var increases by 1 unit, we predict y var  will increase/dec  by ____  y var units.

For y-intercept: When x var = 0 units, we predict that the y var  will be ____ units..

For SLR: Error = epsilon = y - yhat = y - (betahat0 + betahat1x)

SSE = residual1^2 + res. 2^2 +…+ res.  n^2

Standard error of regression = Root MSE (in SAS language)

The text lists six conditions for simple linear... Continue reading "Understanding Simple Linear Regression: R-squared, Slope, and Conditions" »

1. Integration of Polynomials

Let f(x) = -x² + 2x + 3. The integral is:

• a) ∫(-x² + 2x + 3) dx = -x³/3 + x² + 3x + C
• b) [-x³/3 + x² + 3x] from 1 to 2 = 11/3

2. Finding f(x) from f'(x)

Let f'(x) = 3x² - 3. Given f(2) = 1, find f(x):

f(x) = x³ - 3x + C. Substituting f(2) = 1: 8 - 6 + C = 1, so C = -1. Thus, f(x) = x³ - 3x - 1.

3. Solving for f(x) with Initial Conditions

Let f'(x) = 6x² + 2x - 1. Given f(2) = 5:

f(x) = 2x³ + x² - x + C. Substituting f(2) = 5: 2(8) + 4 - 2 + C = 5, so 18 + C = 5, C = -13. Thus, f(x) = 2x³ + x² - x - 13.

4. Polynomial Integration

Let f(x) = 3x² - 4x:

• a) ∫f(x) dx = x³ - 2x² + C
• b) Evaluation results in 32.

5. Definite Integral Properties

Consider ∫₁⁵ f(x) dx = 6:

• a) ∫₁⁵ 2f(x) dx = 2 ∫₁⁵ f(x) dx = 2(6) = 12
• b) ∫₁³ 3 dx

Key Principles of Financial Budgeting

Entity Principle

Consider the company as a functional entity.

Leadership Principle

The Financial Planning Manager develops and coordinates the budget.

Authority Principle

The Finance Manager supervises and presents the budget to the Board of Directors (Partners).

Participation Principle

Involve all key participants in the budget's preparation.

Commitment Principle

All managers undertake to follow the budget, notifying any deviation in a timely manner.

Goal Principle

The budget is based on the strategic planning objectives.

Accounting Principle

A budget system should mirror the current accounting system.

Measurement Principle

All estimates must be in currency units.

Predictability Principle

The predictions generated must... Continue reading "Financial Budgeting: Principles and Model Development" »

Agent Environments and PEAS

PEAS: Performance, Environment, Actuators, Sensors

Environment Properties

• Fully vs. Partially Observable: Full information or hidden information?
• Deterministic vs. Stochastic: Outcome is known or probability is known (if probability is unknown, it is Nondeterministic).
• Episodic vs. Sequential: Independent actions vs. sequential (dependent) actions.
• Static vs. Dynamic: No environment change vs. environment change over time.
• Semidynamic: The environment does not change with time, but the agent's performance score does.
• Discrete vs. Continuous: Distinct chunks (e.g., Chess) or smooth/measurable values (e.g., Driving).
• Known vs. Unknown: Are the rules of the world given or must they be learned?
• Single vs. Multi-agent: One agent

Lease Accounting: Vehicle Leases and Journal Entries

Today is May 31, X1, and we lease three vehicles to visit our factories for €120,000. The contractual conditions are as follows:

• The term of the lease is 4 years.
• The payments (€35,000 each) are annual and will be made on May 31. An additional €1,000 will be added as a purchase option in the final payment.
• Effective interest rate: 6.75%
• The estimated useful life for each vehicle is 8 years.

Lease Amortization Schedule

Journal Entries for Lease Transactions

May 31, X1: Initial Lease Recognition