Statistics Lecture - Tuesday, July 2

Announcements

• No class on Thursday (Fourth of July).
• Fourth exam covers up to what is finished by tomorrow.
• Behind on video editing, shifting to quicker edits.

Recap of Chapter 11, Section 5 - Expected Values

Definition & Formula

• Expected Values: Useful for evaluating investment opportunities
  • Consists of possible outcomes with corresponding probabilities.
  • Sum of probabilities equals 1.
  • Formula: E(X) = X1 * P(X1) + X2 * P(X2) + ... + Xn * P(Xn)*

Application: Investment Opportunities

• Example: Mark investing $6,000 in two companies.
  • Company ABC: Outcomes -400 (0.2), 800 (0.5), 1300 (0.3)
  • Company PDQ: Outcomes 600 (0.8), 1000 (0.2)
  • Expected Profits: ABC = $710, PDQ = $680
  • Suggestion: Invest in Company ABC.

Business & Insurance Applications

• Example: Lumber wholesaler purchase & expected profit.
  • Resell possibilities: $99,500 (0.25), $99,000 (0.60), $88,500 (0.15)
  • Expected Revenue: $95,900
  • Profit calculation: Revenue minus cost; set profit >= $2,500.
  • Conclusion: Pay up to $63,450 to ensure expected profit.

Chapter 12, Section 1 - Simulation Methods

Definition

• Simulation: Mimicking phenomena using simpler probabilities.
• Monte Carlo Methods: Use large numbers of random digits, often via computers.

Application of Simulation

• Example: Genetic probabilities using coin toss.
  • Dominant (R) vs. recessive (r) allele.
  • Procedure: Toss 2 coins 50 times, approximate outcomes.
  • Experimental: 20 out of 48 sequences were RRR. Probability = 20/48 = 0.417.

Example: Estimating Probability with Simulation

• Problem: Probability of having more than 3 boys in a 5-child family using random digits.
  • Representation: Odd digits = boys, even digits = girls.
  • Results: 10 out of 50 families had >3 boys. Probability = 10/50 = 0.2

Chapter 12: Statistics - Introduction

Basic Concepts

• Population: All items of interest.
• Sample: Subset of population.
• Raw Data: Unorganized data (quantitative or qualitative).
• Descriptive Statistics: Collecting, organizing, summarizing data.
• Inferential Statistics: Drawing conclusions about a population based on sample data.

Graphical Representation

• Histograms: Frequency distribution for quantitative data. Bars touch each other.
• Bar Graphs: Similar to histograms but for categorical data and bars do not touch.
• Circle Graphs/Pie Charts: Sectors represent proportional parts of a whole.
• Line Graphs: Show trends over time.
• Stem-and-Leaf Displays: Quick way to preserve data and visually display frequency.

Summary

• Importance of understanding measures of central tendency (mean, median, mode).
• Difference between populations and samples.
• Utilizing graphs to visually interpret data.