Probability Rules and Vocabulary

Basic Probability Concepts

Probability is the study of randomness and uncertainty, providing a mathematical framework for quantifying the likelihood of events. Understanding probability rules and terminology is foundational for statistical inference.

• Event: A specific outcome or set of outcomes from a random process.

• Sample Space (S): The set of all possible outcomes.

• Complement: The set of outcomes in the sample space that are not in the event A, denoted as A'.

• Mutually Exclusive Events: Events that cannot occur at the same time (no overlap).

• Independent Events: The occurrence of one event does not affect the probability of the other.

• Dependent Events: The occurrence of one event affects the probability of the other.

• Conditional Probability: The probability of event A given that event B has occurred, denoted as P(A|B).

Probability Rules

• Complement Rule:

• Addition Rule (for mutually exclusive events):

• Addition Rule (general):

• Multiplication Rule (independent events):

• Conditional Probability:

Example: If P(A) = 0.3 and P(B) = 0.5, and A and B are independent, then P(A and B) = 0.3 × 0.5 = 0.15.

Normal Distribution

Properties and Calculations

The normal distribution is a symmetric, bell-shaped distribution characterized by its mean (μ) and standard deviation (σ). Many natural phenomena follow a normal distribution.

• Standardization (z-score):

• Finding x from z:

• Empirical Rule (68-95-99.7 Rule):

  • About 68% of data falls within 1σ of the mean

  • About 95% within 2σ

  • About 99.7% within 3σ

• TI-84 Functions: NormalCDF (find area/probability), InvNorm (find value for a given probability)

Example: If μ = 100, σ = 15, what is the z-score for x = 130?

Central Limit Theorem (CLT)

Sampling Distributions

The Central Limit Theorem states that the sampling distribution of the sample mean (or proportion) approaches a normal distribution as the sample size increases, regardless of the population's distribution, provided certain conditions are met.

• For Means:

  • Mean:

  • Standard Error (SE):

• For Proportions:

  • Mean:

  • Standard Error:

• Conditions: Random sample, independence, and success/failure (np ≥ 10, n(1-p) ≥ 10)

Example: For n = 100, p = 0.4,

Confidence Intervals

Estimating Population Parameters

Confidence intervals provide a range of plausible values for a population parameter, based on sample data.

• General Form: Estimate ± Margin of Error (ME)

• Margin of Error:

• For Proportions:

• For Means: (use z if σ known)

• Common Critical Values:

  • 90%: 1.645

  • 95%: 1.96

  • 99%: 2.576

Example: For a sample proportion of 0.6, n = 100, 95% CI:

One-Proportion Hypothesis Test

Testing Population Proportions

Hypothesis testing for a single proportion involves comparing the observed sample proportion to a hypothesized value.

• Null Hypothesis (H₀):

• Alternative Hypothesis (Hₐ): , , or

• Test Statistic:

• Decision Rule: Reject H₀ if p-value ≤ α (significance level)

Example: If , , ,

Two-Proportion Inference

Comparing Two Population Proportions

Inference for two proportions involves constructing confidence intervals or hypothesis tests to compare two population proportions.

• Pooled Proportion (for tests):

• Confidence Interval:

• Hypothesis Test: Use pooled proportion in standard error calculation.

Example: If , , , , , ,

Errors in Hypothesis Testing

Type I and Type II Errors

Understanding errors in hypothesis testing is crucial for interpreting results.

• Type I Error (α): Rejecting a true null hypothesis.

• Type II Error (β): Failing to reject a false null hypothesis.

Practical vs Statistical Significance: Statistical significance means the result is unlikely due to chance, while practical significance considers whether the effect size is large enough to be meaningful in context.

Sample Size Effects

Impact on Inference

• Larger sample size leads to smaller standard error (SE).

• Smaller margin of error (ME).

• Narrower confidence intervals (CI).

• Easier to reject H₀ if an effect exists.

TI-84 Calculator Functions

Common Statistical Functions

• NormalCDF: Computes probability for a normal interval.

• InvNorm: Finds value corresponding to a given cumulative probability.

• 1-PropZTest: Hypothesis test for one proportion.

• 1-PropZInt: Confidence interval for one proportion.

• 2-PropZTest: Hypothesis test for two proportions.

• 2-PropZInt: Confidence interval for two proportions.

Key Formulas Summary