Midterm #2 Review Guide: Probability, Distributions, Sampling, and Inference

Probability Distributions and the Normal Model

Probability Distributions

Probability distributions describe how the probabilities are distributed over the values of a random variable. They are fundamental in modeling random events and making statistical inferences.

• Probability Distribution: A function that assigns probabilities to each possible value of a random variable.

• Empirical Rule: For a normal distribution, approximately 68% of data falls within one standard deviation, 95% within two, and 99.7% within three standard deviations of the mean.

• Normal Distribution: A continuous, symmetric, bell-shaped distribution characterized by its mean ($\mu$) and standard deviation ($\sigma$).

• Standard Normal Distribution: A normal distribution with mean 0 and standard deviation 1. Any normal variable $X$ can be standardized using $Z = \frac{X - \mu}{\sigma}$.

Example: If heights of adult males are normally distributed with mean 70 inches and standard deviation 3 inches, the probability that a randomly selected male is taller than 73 inches can be found using the standard normal distribution.

Binomial and Bernoulli Distributions

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Bernoulli Trial: An experiment with exactly two possible outcomes: "success" or "failure".

• Binomial Distribution: The probability of observing $k$ successes in $n$ independent Bernoulli trials with probability $p$ of success is given by:

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

• Mean (Expected Value): $\mu = np$

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

Example: Flipping a fair coin 10 times, the probability of getting exactly 6 heads is calculated using the binomial formula with $n=10$, $k=6$, $p=0.5$.

Sampling, Populations, and Statistics

Populations, Samples, Parameters, and Statistics

Understanding the distinction between populations and samples is crucial in statistics.

• Population: The entire group of individuals or items of interest.

• Sample: A subset of the population selected for analysis.

• Parameter: A numerical summary describing a characteristic of the population (e.g., population mean $\mu$, population proportion $p$).

• Statistic: A numerical summary describing a characteristic of a sample (e.g., sample mean $\bar{x}$, sample proportion $\hat{p}$).

Example: If you survey 100 students (sample) from a university (population) and find that 60% prefer online classes, 60% is the sample statistic, while the true proportion for all students is the population parameter.

Surveys, Sampling, and Bias

Proper sampling methods are essential to obtain representative data and valid inferences.

• Measurement Bias: Systematic error due to faulty measurement instruments or procedures.

• Nonresponse Bias: Bias introduced when a significant portion of the sampled individuals do not respond.

• Bias in Surveys: Any systematic error that results in an incorrect estimate of the population parameter.

Example: If a survey about exercise habits is conducted at a gym, the results may be biased toward more active individuals.

Sampling Distributions and the Central Limit Theorem

Sampling Distributions

The sampling distribution describes the distribution of a statistic (like the sample mean or proportion) over many samples drawn from the same population.

• Standard Error (SE): The standard deviation of a sampling distribution. For sample proportion $\hat{p}$:

$SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$

• Central Limit Theorem (CLT): For large sample sizes, the sampling distribution of the sample mean or proportion is approximately normal, regardless of the population's distribution.

Example: If the population proportion is $p=0.4$ and $n=100$, then $SE_{\hat{p}} = \sqrt{\frac{0.4 \times 0.6}{100}} = 0.049$.

Confidence Intervals and Margin of Error

Confidence Intervals for Proportions and Means

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

• Confidence Interval for a Proportion:

$\hat{p} \pm z^* \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$

• Confidence Interval for a Mean (when $\sigma$ is known):

$\bar{x} \pm z^* \frac{\sigma}{\sqrt{n}}$

• Margin of Error (ME): The maximum expected difference between the true population parameter and a sample estimate, given by the product of the critical value and the standard error.

Example: For a sample proportion $\hat{p}=0.6$, $n=100$, and $z^*=1.96$ (for 95% confidence), the margin of error is $1.96 \times \sqrt{0.6 \times 0.4 / 100} = 0.096$.

Hypothesis Testing for Population Proportions

Hypothesis Testing Concepts

Hypothesis testing is a formal procedure for comparing observed data with a claim (hypothesis) about a population parameter.

• Null Hypothesis ($H_0$): The default assumption or claim to be tested (e.g., $p = p_0$).

• Alternative Hypothesis ($H_a$): The competing claim (e.g., $p \neq p_0$, $p > p_0$, or $p < p_0$).

• Test Statistic: A standardized value used to decide whether to reject $H_0$.

• p-value: The probability, under $H_0$, of obtaining a result as extreme or more extreme than the observed result.

• Significance Level ($\alpha$): The threshold for rejecting $H_0$ (commonly 0.05).

Example: Testing if the proportion of defective items is greater than 5% ($H_0: p = 0.05$, $H_a: p > 0.05$).

One-Proportion z-Test

The one-proportion z-test is used to test hypotheses about a single population proportion.

• Test Statistic Formula:

$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

• Compare the p-value to $\alpha$ to decide whether to reject $H_0$.

Example: If $\hat{p}=0.08$, $p_0=0.05$, $n=200$, then $z = \frac{0.08 - 0.05}{\sqrt{0.05 \times 0.95 / 200}}$.

Two-Sample Proportion Test

This test compares the proportions from two independent samples to determine if there is a significant difference between them.

• Test Statistic Formula:

$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

where $\hat{p}$ is the pooled sample proportion.

Central Limit Theorem for Sample Means

The CLT states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution, provided the samples are independent and identically distributed.

• Standard Error for Sample Mean:

$SE_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$

Example: For a population with $\sigma=10$ and $n=25$, $SE_{\bar{x}} = 10/5 = 2$.

Summary Table: Key Concepts and Formulas

Concept	Definition	Key Formula
Normal Distribution	Continuous, symmetric, bell-shaped distribution	$Z = \frac{X - \mu}{\sigma}$
Binomial Distribution	Discrete distribution for number of successes in $n$ trials	$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}$
Standard Error (Proportion)	SD of sample proportion's distribution	$SE_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$
Confidence Interval (Proportion)	Range for population proportion	$\hat{p} \pm z^* SE_{\hat{p}}$
One-Proportion z-Test	Test for single population proportion	$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
Central Limit Theorem	Sampling distribution of mean/proportion is normal for large $n$	---

Additional info: Some formulas and explanations have been expanded for clarity and completeness, based on standard introductory statistics curriculum.