STAT 1000 - Probability & Sampling Distribution of the Sample Mean

STAT 1000 - Basic Statistics Analysis I

Unit 05 - Probability & Sampling Distribution of the Sample Mean

Outline

• Randomness, definition of probability

• Sample space

• Basic probability rules

• Probability distributions

• Sampling distribution of a sample mean

• Central Limit Theorem

Variability and Randomness

• Statistics engages with the concept of variability.

• Key Idea:

  • Random behavior is unpredictable in the short run.

  • However, it exhibits a predictable distribution in the long run.

Probability

• Conceptual Scenario:

  • When rolling a die, tossing a coin, or buying a lottery ticket with chosen numbers, individual outcomes are uncertain.

  • Yet these outcomes can be described by a regular pattern that emerges over numerous trials, laying the foundation for probability studies.

Probability and Fair Coin

• When tossing a fair coin:

  • The likelihood of a Head (H) and a Tail (T) is equal.

  • Probability (P) of either outcome:
    P(H) = P(T) = 0.5

  • Example Sequence: H T T H T H

  • Observed proportions: 1.0, 0.5, 0.33, 0.5, 0.4, 0.5

  • Early fluctuations observed in proportions settle closer to 0.5 in the long run.

  • Example Prediction: Tossing a fair coin 10,000 times, observation will likely fall between 4,900 and 5,100 Heads.

Randomness Defined

• A phenomenon is termed random if individual outcomes are uncertain, yet there is a consistent distribution of outcomes across many repetitions.

• Definition of a Random Experiment:

  • Any process with uncertainty that has two or more possible outcomes.

Difference Between Proportions and Probability

• Proportion: A known or observed value.

• Probability: A theoretical value related to the proportion over an infinitely long series of trials.

• Usage in Language:

  • Proportions are present tense; probabilities relate to potential future events.

Probability Theory

• Defined as the mathematical branch that describes random behavior.

• Note: Probability is not directly observable; mathematical models are essential for analysis.

Probability Model

• Definition of a Probability Model:

  • A framework involving two components:

  • A list of possible outcomes

  • A probability for each outcome

Sample Space

• Sample Space (S): The set of all possible outcomes for a random phenomenon.

  • Example 1: Coin toss, S = {H, T}.

  • Example 2: Lotto 6/49 (population 49), S has nearly 14 million combinations.

  • Example 3: Coin tossed three times, S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

  • Example 4: Quality control with three fuses: S = {NNN, DNN, NDN, NND, DDN, DND, NDD, DDD}.

  • Number of outcomes calculated as: 2^n, where n is the number of coins or fuses examined.

Probabilities of Outcomes

• For outcomes O1, O2, …, On within sample space S:

  • Denote probabilities as pi for outcome Oi.

  • Conditions for probabilities:

  • 0 \leq p_i \leq 1 for each i = 1, 2, …, n.

  • p1 + p2 + … + p_n = 1

Events

• Definition: An event is any subset of outcomes from the sample space.

  • Example: Flipping a coin three times for exactly two Tails, A = {HTT, THT, TTH}.

  • Example: Rolling two dice and summing to 9, B = {36, 45, 54, 63}.

Complements of Events

• Complement of event A (denoted A^c): Set of all outcomes not contained in A.

  • Example 1: Flipping a coin, H^c = T.

  • Example 2: Flipping three coins where A = {exactly two tails}, A^c = {HHH, HHT, HTH, THH, TTT}.

• Relation:

  • The sum of probabilities follows: P(A^c) = 1 - P(A)

  • Example Usage: P(Bombers win) = 1 - P(Bombers lose)

Random Variables

• Definition: A random variable (RV) is a numerical outcome of a random phenomenon.

• Types of Random Variables:

  • Discrete: Countable outcomes (e.g., number of children in a family).

  • Continuous: Uncountable outcomes within an interval (e.g., weight, distance).

Probability Distributions

• A probability distribution maps variable values to probabilities.

• Example Distribution: Rolling a fair six-sided die, let X be the face showing.

• Case Study: Dice painted in multiple colors with given probabilities for rolling a color.

Probability of Events

• Method for calculating the event probability involves summing probabilities of contained outcomes.

• Example: Flipping three coins for two tails, A = {HTT, THT, TTH}:

  • Calculation:
    P(A) = P(HTT)+P(THT)+P(TTH)

Probability Distributions of Sums

• Example: When rolling two dice, the outcome X as the sum:

  • To find P(X > 8), the expression is:
    P(X > 8) = P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)
    = \frac{4}{36} + \frac{3}{36} + \frac{2}{36} + \frac{1}{36} = \frac{10}{36} = 0.2778

NHL Example

• Atlantic Division teams each have calculated probabilities for winning.

  • The sum of calculated probabilities must equal 1.

  • To find the probability of a Canadian team winning:
    P(Canadian team wins) = P(Montreal wins) + P(Ottawa wins) + P(Toronto wins) = \text{k} + 0.02 + 4\text{k}

  • Solving for k yields:

  • Hence,
    P(Canadian team wins) = 0.37

American Team Winning Probability

• For the Pacific Division teams, to find the probability for an American team:

  • Calculate as follows:
    P(American team wins) = 1 - P(Canadian team wins)

Probabilities for Continuous Variables

• Dealing with continuous random outcomes requires approaching by intervals of values rather than single points.

• Assigning probabilities by assessing areas under density curves represents chances in given intervals.

Normal Distribution Example

• Example scenario: Pulse rates of adult females in normal distribution, mean 74 bpm, std dev 12 bpm.

  • Calculation required for selecting above certain thresholds.

Distribution of the Sample Mean

• Exploring mean calculations for random samples from populations.

• Process involves taking several samples of size n, calculating their mean, and plotting results to observe distribution.

  • Example: Heights of adult Canadian males normally distributed as X ~ N(178, 6).

Sampling Distribution of Sample Mean

• For sample mean calculations:

  • Essential rule: if X is random variable with mean μ and deviation σ, then
    \bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})

• Conditions:

  • Distribution becomes normal as sample size increases (the Central Limit Theorem).

Central Limit Theorem

• Fundamental statistical theorem stating:

  • For large sample sizes (n), the sampling distribution of the mean will approximate normal distribution regardless of the population distribution.

• Implication of the sample size: Greater the sample, faster convergence to normal distribution.

  • Often applied with n ≥ 30 as a threshold for safety in statistical assumptions.

Example - Light Bulb Lifetimes

• Light bulb lifetime exhibiting a right-skewed distribution, mean 400 hours, std dev 250 hours.

  • Evaluating probabilities from selected samples of light bulbs, even without initial normality in underlying distribution due to applying CLT for large samples.

Summary of Sampling Distribution

• When X follows a normal distribution,

  • Sampling distribution reflects directly as normal.

• For non-normal distributions, yields approximately normal as long as n is significantly large.