Notes, summaries, assignments, exams, and exercises for Mathematics for Other courses

Hypothesis Testing

Statistical Test Selection

1. If the population standard deviation is unknown and the sample size is less than 30: t-test

2. If the population standard deviation is known and the sample size is less than 30: t-test

7. Hypothesis test on population mean; n = 25; σ = 2.5: z-test

8. Hypothesis test on population mean; n = 50; s = 7.2: z-test

18. Test statistic for sample size above 30: z-test

19. Test statistic when population standard deviation is known: z-test

20. Test statistic when population standard deviation is unknown: t-test

21. When to use the t-test: I and II

24. Optimal sample size for z-test: Equal to or larger than 30

Hypotheses and Significance

3. H₀: μ = 30

4. H₁: μ > 30

5. No

9. False: The alternative hypothesis typically...

Research Concepts and Terminologies

• Element: The basic unit of the population.
• Population: The entire group of interest.
• Study Population: The specific subset being studied.
• Sample: A subset of the population.
• Sampling Unit: The individual entity selected.
• Sampling Frame: The list of all units in the population.
• Observation Units: The units from which data is collected.
• Variable: A characteristic that varies.
• Parameter: A numerical characteristic of a population.
• Statistics: Numerical characteristics of a sample.
• Sampling Error: The difference between sample and population values.

Sampling Procedures

Non-Probability Sampling

• Accidental Sampling
• Quota Sampling
• Convenience Sampling
• Purposive Sampling

Probability Sampling

• Random Sampling
• Systematic Sampling
• Stratified

Correlation and Statistical Indices

• Positive Correlation: As one variable increases, the other also increases.
• Negative Correlation: As one variable increases, the other decreases.
• Correlation Coefficient: A statistical index of the relationship between two things, ranging from -1 to +1.

Measures of Central Tendency

A statistical measure that identifies a single value as representative of an entire distribution.

• Mean: The average of a set of numerical values.
• Median: The middle value in a list of numbers.
• Mode: The value that appears most often in a set of data.

Variation and Distribution

• Range: The difference between the highest and lowest values in a dataset.
• Standard Deviation: A measure of the amount of variation or dispersion in a set of values.
• Regression

(a) Alphabet, String, and Language Definitions

• Alphabet (Σ): A finite, non-empty set of symbols. Example: Σ = {0, 1} or Σ = {a, b}.
• String: A finite sequence of symbols chosen from an alphabet. Example: 0110, abba.
• Language (L): A set of strings over an alphabet Σ. Example: L = {a, ab, aab, aaab, ...} = a⁺b over Σ = {a, b}. ε denotes the empty string (length 0).

(b) DFA vs. NFA

• DFA: For each state and input symbol, exactly one next state is defined. No ε-transitions are allowed.
• NFA: For each state and input symbol, zero, one, or more next states are possible. ε-transitions are allowed.
• Power: Both accept the same class of languages (Regular Languages). Every NFA can be converted to an equivalent DFA.
• Design: DFA usually has more states;

Matrices and Determinants

• Inverse Matrix: |A|⁻¹ = 1/|A|; A⁻¹ = (Adj A)ᵀ / |A|; A⁻¹ · A = I.
• Multiplication: Row by column (fila por columna).
• Determinants: |A₃,₃| = 4, det(2A) = 2³ · 4.
• Resolution (Cramer's Rule): x = |B c₂ c₃| / |A|, y = |c₁ B c₃| / |A|...

Geometry and Spatial Relationships

Areas and Volumes

• Area of a Parallelogram: |a × b|
• Area of a Triangle: |a × b| / 2
• Volume of a Parallelepiped: [u, v, w]
• Volume of a Tetrahedron: [u, v, w] / 6

Relative Positions

• Position of 3 Planes:
  • rg(A) = rg(A*) = 3: System Consistent Determined (1 point).
  • rg(A) = rg(A*) = 2: Line (recta).
  • rg(A) = rg(A*) = 1: Coincident.
  • rg(A) = 1, rg(A*) = 2: Parallel.
  • rg(A) = 2, rg(A*) = 3: Two cases: if two rows are proportional, two planes are parallel

Understanding Cluster Analysis

Cluster analysis is a statistical method used to process data by organizing items into groups (clusters) based on their associations. The objective is to identify similar groups of subjects where shared features indicate a global set of characteristics. This method is employed when there are no prior assumptions regarding the relationships within the data. Subjects are separated into groups so that each subject is more similar to others within its group than to those outside of it (e.g., age, income, location).

Key Metrics

• Intra-cluster distance: The distance between data points inside a cluster. The goal is to minimize this distance so observations are as similar as possible.
• Inter-cluster distance: The distance

Variables in Didactic Rankings

The didactic nature of the attributes or properties involves the degree of abstraction of the material, the possible conjunction or disjunction of attributes, the use of denial, the size of the collection, and the number of property values.

Key Milestones in Measurement

Consideration and Magnitude Perception

One must consider and perceive magnitudes; there are quantities that are very obvious, such as length. Regarding weight (mass), to perceive and consider it, children can perform activities such as weighing with their hands or using a balance. They often choose "higher" even if it is within.

Conservation of Magnitude

If I have a ball of dough and squash it, it changes shape, but not weight.

Magnitude Management and

Operations Research

1. Operations Research
Operations Research (OR) is a scientific and mathematical approach used to help managers make the best possible decisions under given constraints and limited resources.

Scope

• Production – production planning and product mix.
• Marketing – advertising and sales decisions.
• Finance – investment and budgeting.
• HR – manpower planning and scheduling.
• Transportation – reducing transportation cost.
• Inventory – managing stock and resources.
• Project Management – planning and scheduling.

Limitations

• Depends on accurate data.
• Mathematical models may be complex.
• Models are based on assumptions that may not reflect reality.
• Human behaviour is difficult to express mathematically.
• Requires skilled people and technology.

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...

Cost, Revenue, and Demand Calculations

Variable Expenses Formula:
Variable Expenses = $0.75 per Donut × Number of Donuts Produced

Example: If 285.00 units are made and demand is 261.00, your fixed expenses are $1,257.00, your variable cost per unit is $2.25, and your sales price is $6.67. What are the variable expenses for the day?
Calculation: 285.00 × $2.25 = $641.25.

Revenue Formula:
Revenue = Selling Price per Donut × Donuts Sold

Example: If 294.00 units are made and demand is 245.00, your variable cost per unit is $2.25, and your sales price is $6.27. What is the revenue for the day?
Calculation: 245 × $6.27 = $1,536.15.

Inventory Discrepancy Definitions

• Unhappy Customer Expense: Demand is greater than production.
• Leftovers: Demand is less than