Probability and Sampling Methods Quiz

Data and Decisions - Probability and Sampling Distributions

• Unit 1: Probability and Sampling Distributions
• Session 01: Topics covered include Discrete and Continuous Probability Distributions, Normal Probability Distribution, Descriptive Statistics and Inference, Measurement Scales, Data Collection, Data Visualization and Sampling Distributions
• Topics Covered (Page 2):
  • Discrete and Continuous Probability Distributions
  • Continuous Probability Distributions (Normal Distribution)
  • Descriptive & inferential statistics
  • Measurement scales, data collection, data visualization
  • Sampling Distributions
  • Sampling Methods: Simple Random Sampling, Stratified Random Sampling, Cluster Sampling, Systematic Sampling, Convenience Sampling, Judgement Sampling. Two Case Studies
• Random Experiments (Page 4): Experiments where the value of certain variables cannot be controlled, and results vary from one performance to the next, even when most conditions remain the same.
• Examples of Random Experiments (Page 4-5):
  • Tossing a coin
  • Tossing a die
  • Manufacturing bolts (defective/non-defective)
  • Measuring lifetimes of electric light bulbs.
• Sample Spaces (Page 6-7): A set of all possible outcomes of a random experiment.
  • Finite, countably infinite, or uncountably infinite sample spaces.
• Events (Page 8-9): Subsets of the sample space.
  • Simple or elementary event consists of a single sample point.
  • Impossible (empty set) or sure (entire sample space) event.
  • Union (A∪B), Intersection (A∩B), Complement (A'), and difference (A − B)
• Mutually Exclusive Events: Two events A and B are mutually exclusive if A∩B= Φ.
• Example 1.9 (Page 11): Experiment of tossing a coin twice, with events A (at least one head) and B (second toss is a tail). Describes Set Operations on events.
• The Concept of Probability (Page 12): A measure of the chance of an event occurring, between 0 and 1.
  • Classical approach (equally likely outcomes: h/n)
  • Frequency approach (empirical probability: h/n)
• The Axioms of Probability (Page 13):
  • 0 ≤ P(A) ≤ 1
  • P(S) = 1; S is the sample space.
  • P(A U B) = P(A) + P(B) if A and B are mutually exclusive
  • Generalized P(A U B) = P(A) + P (B) - P(A∩B)
• Example 1.12 (Page 14): probability of rolling a 2 or 5 on a single die.
• Example 1.13 (Page 14): probability of a number less than 4 if no other information is given or if we know the toss resulted in an odd number.
• Types of Events (Page 15-16): Simple events have one sample result (e.g., 'heads' on a coin toss), while compound events have more than one (e.g., 'at least one head').
• Algebra of Events (Page 17-18): Union (A∪B), Intersection (A∩B), Complement (A'), and differences (e.g., A-B = A∩B’)
• Probability and sampling Distributions (pages 20-24): examples calculations. (a-g) Examples on finding the probability that a card drawn is an ace, a jack of hearts, a three of clubs, or six of diamonds, a heart, any suit except hearts, etc.
• Sampling Problems using given data (pages 24-31). Problems involving boxes of colored marbles, finding the probability a red marble is chosen when from a particular box is chosen.
• Random Variables (Page 32-35): A function or variable assigned to each point in a sample space that assigns numbers.
  • Discrete random variable (finite or countably infinite set of values)
  • Probability function (sum of probabilities equals 1)
• Variables and Distribution functions Example 1.11 and 1.11 (pages 33-38). Example on finding the probability function from random variables.
• Mathematical Expectation (Page 39-41): The average output for a random variable, often the expected value.
• Variance and Standard Deviation (Page 41-43): Measures of dispersion or scatter of values from the mean.
• Properties of Binomial Distribution (Page 49): Formulas for the mean, variance, and standard deviation (μ=np, σ² = npq, σ=√npq)
• The Poisson Distribution (Pages 50-57): A discrete distribution describing the probability of a given number of events occurring in a fixed interval of time (or space) if these events occur with a known average rate and independently of the time since the last event. (Poisson formula calculations and problems).
• The Normal Distribution (Pages 58-62): A continuous probability distribution that is symmetrical, bell-shaped, and asymptotic (never touches the x-axis).
• Areas under the Normal Curve (Pages 61-76): Calculating Probabilities from Z-scores showing how to find the areas in a specific interval with given μ and σ.
• Sampling Distributions (Pages 77-84): Distributions of sample statistics.
  • Random Sampling Methods: Simple Random Sampling, Systematic Sampling, Stratified Sampling, Cluster Sampling.
  • Non-random Sampling Methods: Convenience Sampling, Judgment Sampling, Quota Sampling, Snowball Sampling.
• Hypothesis Testing (Pages 85-84): A procedure to determine if a particular condition about a population is true based on a sample.
  • Null hypotheses (the assumed state of being true)
  • Alternative Hypothesis (the proposed alternative)
  • Types of errors (Type 1 - rejecting a true null, Type 2 – accepting a false null)
  • Level of significance (α) and confidence level (1-α).
  • t-tests (single or paired samples compared against values using the standard deviation of the calculated means with given data)
  • χ² tests for categorical variable relationship.