STAT 2 Review: Probability, Distributions, and Estimation

STAT 2 Review

Exam Information and Allowed Materials

This review covers key topics for an upcoming statistics exam. Students are permitted to bring a single, handwritten 8.5"x11" formula/note sheet (no examples allowed) and a calculator. Sharing devices is not permitted during the exam.

Key Terminology

• Sample space: The set of all possible outcomes in a probability experiment.

• Event: Any subset of a sample space.

• Mutually exclusive: Events that cannot occur at the same time.

• Random variable: A variable that assigns a numerical value to each outcome in a sample space.

• Discrete random variable: Takes on a countable number of values.

• Continuous random variable: Takes on an infinite number of possible values within a range.

• Probability distribution: Describes how probabilities are distributed over the values of the random variable.

• Probability histogram: A graphical representation of a probability distribution for a discrete random variable.

• Bernoulli trials: Experiments with exactly two possible outcomes (success or failure).

• Binomial distribution: The probability distribution of the number of successes in a fixed number of independent Bernoulli trials.

• Density curve: A curve that describes the overall pattern of a continuous probability distribution.

• Normal distribution: A symmetric, bell-shaped, continuous probability distribution defined by its mean and standard deviation.

• Central Limit Theorem: States that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases.

• Point estimate: A single value estimate for a population parameter (e.g., sample mean for population mean).

• Confidence interval: A range of values used to estimate a population parameter.

• Confidence level: The probability that the confidence interval contains the true population parameter.

• Margin of error: The maximum expected difference between the point estimate and the true population parameter.

Probability and Counting Principles

Probability for Equally Likely Outcomes

• For an event E in a sample space with n equally likely outcomes: Probability of E: $P(E) = \frac{\text{Number of outcomes in } E}{\text{Total number of outcomes}}$

Compound Events

• Complement: $P(\text{not } E) = 1 - P(E)$

• Intersection (A and B): Probability both A and B occur.

• Union (A or B): Probability at least one of A or B occurs.

Probability Notation and Rules

• Addition Rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

• Complement Rule: $P(A) = 1 - P(\text{not } A)$

Counting Principles

• Multiplication Principle: If one event can occur in m ways and a second in n ways, the two events together can occur in $m \times n$ ways.

• Permutations: Number of ways to arrange n objects taken r at a time (order matters, no repetition): $nP_r = \frac{n!}{(n-r)!}$

• Combinations: Number of ways to choose r objects from n (order does not matter, no repetition): $nC_r = \frac{n!}{r!(n-r)!}$

Random Variables and Probability Distributions

Discrete Random Variables

• Mean (Expected Value): $\mu = \sum x \cdot P(X = x)$

• Standard Deviation: $\sigma = \sqrt{\sum x^2 \cdot P(X = x) - \mu^2}$

Binomial Distribution

• Describes the number of successes in n independent Bernoulli trials with probability p of success.

• Mean: $\mu = np$

• Standard Deviation: $\sigma = \sqrt{np(1-p)}$

• Sampling Without Replacement: If the sample size is less than 5% of the population, the binomial distribution can be used as an approximation.

Normal Distribution

• General Normal Distribution: Defined by mean $\mu$ and standard deviation $\sigma$.

• Standard Normal Distribution: $\mu = 0$, $\sigma = 1$

• To find probabilities or values from areas under the curve, use standardization: $z = \frac{x - \mu}{\sigma}$

Sampling Distributions and Central Limit Theorem

• Sampling Distribution of the Mean:

  • Mean: $\mu_{\bar{x}} = \mu$

  • Standard Deviation: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

• Central Limit Theorem: For large n, the distribution of sample means is approximately normal, regardless of the population's distribution.

Estimation: Confidence Intervals for the Mean

When Population Standard Deviation ($\sigma$) is Known

• Point Estimate: $\bar{x}$ (sample mean)

• Margin of Error: $E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

• Confidence Interval: $\bar{x} \pm E$ or $\bar{x} - E$ to $\bar{x} + E$

• Sample Size for Desired Margin of Error: $n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2$ (always round up to the next integer)

When Population Standard Deviation ($\sigma$) is Unknown

• Point Estimate: $\bar{x}$

• Margin of Error: $E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

• Confidence Interval: $\bar{x} \pm E$ or $\bar{x} - E$ to $\bar{x} + E$

• Alternative Margin of Error Calculation: $E = \frac{\text{Upper Limit} - \text{Lower Limit}}{2}$

Statistical Software: StatCrunch Instructions

• For binomial and normal calculations, use StatCrunch's calculator functions.

• For confidence intervals, use StatCrunch's Z Stats or T Stats (One Sample) as appropriate.

Summary Table: Key Formulas

Concept	Formula
Mean of Discrete RV	$\mu = \sum x \cdot P(X = x)$
Std Dev of Discrete RV	$\sigma = \sqrt{\sum x^2 \cdot P(X = x) - \mu^2}$
Mean of Binomial	$\mu = np$
Std Dev of Binomial	$\sigma = \sqrt{np(1-p)}$
Std Dev of Sample Mean	$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$
Margin of Error (Z)	$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$
Margin of Error (t)	$E = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$
Sample Size (Z)	$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2$

Additional info: These notes are structured to provide a comprehensive review for a college-level statistics exam, covering probability, distributions, and estimation. All formulas are provided in LaTeX for clarity and exam preparation.