Study Guide: Probability Distributions, Normal Distributions, and Confidence Intervals (Chapters 4-6)

Probability Distributions

Discrete vs. Continuous Random Variables

Random variables can be classified as either discrete or continuous. Understanding the distinction is fundamental in probability and statistics.

• Discrete Random Variable: Takes on a countable number of distinct values (e.g., number of heads in 10 coin tosses).

• Continuous Random Variable: Can take on any value within a given interval (e.g., height, weight, time).

Example: The number of students in a classroom is discrete; the amount of rainfall in a day is continuous.

Probability Distributions

A probability distribution describes how probabilities are distributed over the values of the random variable.

• For a discrete random variable, the probability distribution lists each possible value and its probability.

• For a continuous random variable, the probability distribution is described by a probability density function (PDF).

Criteria for a Discrete Probability Distribution:

• Each probability is between 0 and 1: $0 \leq P(x) \leq 1$

• The sum of all probabilities is 1: $\sum P(x) = 1$

Reading and Interpreting Discrete Probability Distributions

• Probability distributions can be presented as tables or histograms.

• To find the probability of an event, sum the probabilities of the relevant outcomes.

Example Table:

x	P(x)
0	0.2
1	0.5
2	0.3

Mean, Variance, and Standard Deviation of a Discrete Probability Distribution

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

• Variance: $\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]$

• Standard Deviation: $\sigma = \sqrt{\sigma^2}$

Interpretation: The mean represents the long-run average outcome; the variance and standard deviation measure the spread.

Binomial Experiments and Binomial Probability

A binomial experiment is a specific type of probability experiment with the following properties:

• Fixed number of trials (n)

• Each trial has two possible outcomes: success or failure

• Probability of success (p) is the same for each trial

• Trials are independent

Binomial Probability Formula:

$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$

• binompdf: Used to find the probability of exactly k successes.

• binomcdf: Used to find cumulative probabilities (at least, at most, less than, greater than, between).

Example: Probability of getting exactly 3 heads in 5 coin tosses (p = 0.5):

$P(X = 3) = \binom{5}{3} (0.5)^3 (0.5)^2 = 10 \times 0.125 \times 0.25 = 0.3125$

Normal Probability Distributions

Properties and Interpretation

The normal distribution is a continuous, symmetric, bell-shaped distribution characterized by its mean ($\mu$) and standard deviation ($\sigma$).

• The total area under the curve is 1.

• The mean, median, and mode are all equal and located at the center.

• Empirical Rule: Approximately 68% of data falls within 1$\sigma$, 95% within 2$\sigma$, and 99.7% within 3$\sigma$ of the mean.

Standard Normal Distribution and Z-Scores

The standard normal distribution has a mean of 0 and a standard deviation of 1. A z-score measures how many standard deviations a value is from the mean.

Z-Score Formula:

$z = \frac{x - \mu}{\sigma}$

• Use z-scores to find areas (probabilities) under the standard normal curve using tables or calculators.

• To find the area to the left of a z-score, use the standard normal table.

• To find the area to the right, use the complement rule: $P(Z > z) = 1 - P(Z < z)$

Finding Probabilities and Areas

• normalcdf: Calculator function to find the area between two z-scores or x-values.

• To use the standard normal table, convert x-values to z-scores first.

Example: Find the probability that a value is less than 1.5 standard deviations above the mean:

$P(Z < 1.5)$ (Look up 1.5 in the z-table or use normalcdf)

Finding Z-Scores and X-Values from Areas

• Given an area (probability), find the corresponding z-score using the table or invNorm on the calculator.

• To find the x-value corresponding to a given probability:

$x = \mu + z \sigma$

Sampling Distributions and the Central Limit Theorem

Sampling Distributions

A sampling distribution is the probability distribution of a statistic (like the sample mean) based on all possible samples of a fixed size from a population.

• The mean of the sampling distribution of the sample mean is $\mu_{\bar{x}} = \mu$

• The standard deviation (standard error) is $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Central Limit Theorem (CLT)

The Central Limit Theorem states that, for a sufficiently large sample size (usually n ≥ 30), the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

• Allows use of normal probability methods for sample means.

• Used to find probabilities involving sample means using normalcdf.

Confidence Intervals

Point Estimate and Margin of Error

• Point Estimate: A single value estimate for a population parameter (e.g., sample mean $\bar{x}$ for population mean $\mu$).

• Margin of Error (E): The maximum likely difference between the point estimate and the true parameter.

Margin of Error Formula (for mean, sigma known):

$E = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

Confidence Intervals for a Population Mean (Sigma Known)

• Formula: $\bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}}$

• Use ZInterval on the calculator.

• Find critical z-values from the standard normal table.

Minimum Sample Size for Estimating a Mean

To achieve a desired margin of error E at a given confidence level:

$n = \left( \frac{z_{\alpha/2} \cdot \sigma}{E} \right)^2$

Confidence Intervals for a Population Mean (Sigma Unknown)

• Use the t-distribution when population standard deviation is unknown.

• Formula: $\bar{x} \pm t_{\alpha/2, df} \cdot \frac{s}{\sqrt{n}}$

• Use TInterval on the calculator.

• Find t-values from the t-distribution table or calculator (invT).

Confidence Intervals for a Population Proportion

• Point estimate: $\hat{p} = \frac{x}{n}$

• Formula: $\hat{p} \pm z_{\alpha/2} \cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Use 1-PropZInterval on the calculator.

Minimum Sample Size for Estimating a Proportion

$n = \frac{z_{\alpha/2}^2 \cdot \hat{p}(1-\hat{p})}{E^2}$

Example: To estimate a population proportion with 95% confidence and margin of error 0.05, use the formula above with $z_{\alpha/2} = 1.96$.

Additional info: This guide covers the main concepts and formulas from Chapters 4-6, including probability distributions, the normal distribution, the Central Limit Theorem, and confidence intervals for means and proportions. Calculator functions (binompdf, binomcdf, normalcdf, invNorm, ZInterval, TInterval, 1-PropZInterval) are referenced as per the exam outline.