Probability and Sampling Distributions Cheat Sheet

Understanding probability distributions is crucial for making inferences about populations from sample data. Sampling distributions, particularly the sampling distribution of the mean, are theoretical constructs that describe the variability of sample statistics and are foundational to hypothesis testing.

Core Principles

Probability is the long-run relative frequency of an event, scaled from 0 (impossible) to 1 (certain).
Probability distributions describe the likelihood of different outcomes for a variable.
Discrete variables have distinct, separate values (e.g., number of heads in coin flips).
Continuous variables can take any value within a range (e.g., height, temperature).
For continuous distributions, probability is represented by the area under the curve.
The total area under any probability distribution curve equals 1.
The Normal Distribution (bell curve) is a common continuous distribution where data clusters around the mean.
Sampling distributions describe the distribution of a statistic (like the mean) calculated from repeated samples.
The Central Limit Theorem states that sampling distributions of the mean tend towards normality as sample size increases.
The standard error of the mean quantifies the variability of sample means around the population mean.

Action Steps

Define probability as long-run relative frequency.
Identify whether a variable is discrete or continuous.
Understand that probability in continuous distributions is the area under the curve.
Recognize the properties of the Normal Distribution (e.g., 68-95-99.7 rule).
Differentiate between population parameters and sample statistics.
Apply the Central Limit Theorem to understand the shape of sampling distributions.
Calculate the standard error of the mean to estimate variability.
Use z-scores and probability tables to find probabilities associated with sample statistics.
Relate sampling distributions to hypothesis testing for uncertainty estimation.

Formulas

$z = \frac{y - \bar{y}}{s_y}$
$\sigma_{\bar{y}} \approx \frac{s_y}{\sqrt{n}}$
$s_y^2 = \frac{\sum_{i=1}^{n}(y_i - \bar{y})^2}{n - 1}$

Key Terms

Probability: The likelihood of an event occurring, expressed as a number between 0 and 1.
Probability Distribution: A function that describes the likelihood of obtaining the possible values that a random variable can assume.
Discrete Variable: A variable whose value is obtained by counting.
Continuous Variable: A variable that can take any value within a given range.
Normal Distribution: A symmetric, bell-shaped probability distribution characterized by its mean and standard deviation.
Sampling Distribution: A probability distribution of a statistic obtained through repeated random sampling from a population.
Central Limit Theorem (CLT): States that the sampling distribution of the sample mean approaches a normal distribution as the sample size gets larger, regardless of the population's distribution.
Standard Error of the Mean (SEM): The standard deviation of the sampling distribution of the sample mean; it measures the variability of sample means.
Z-score: A measure of how many standard deviations a data point is from the mean.

Timeline

February 3rd, 2026: Lecture on Probability Distributions and Sampling Distributions.
February 10th: Midterm Exam.
February 24th: Assignment 5 Due; Reading Report 5 Due.
February 27th: Sleep Log 5 Due.

People

Not specified: Instructor/TA