Exam 2 Study Guide: Probability Distributions, Sampling, Confidence Intervals, and Hypothesis Testing

Chapter 7: Probability Distributions

Uniform Distribution

The uniform distribution is a continuous probability distribution where all outcomes in a given interval are equally likely.

• Probability Density Function (PDF): For a uniform distribution on the interval [a, b], the PDF is given by:

$f(x) = \begin{cases} \frac{1}{b-a} & \text{for } a \leq x \leq b \\ 0 & \text{otherwise} \end{cases}$

• Calculating Probabilities: The probability that X falls between c and d (where a ≤ c < d ≤ b) is:

$P(c \leq X \leq d) = \frac{d-c}{b-a}$

• Example: If X ~ Uniform(2, 8), then $P(3 \leq X \leq 5) = \frac{5-3}{8-2} = \frac{2}{6} = 0.333$.

Normal Distribution

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

• Standard Normal Distribution: A normal distribution with mean 0 and standard deviation 1, denoted as Z ~ N(0, 1).

• Z-score: The number of standard deviations a value x is from the mean:

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

• Using the Normal Table: The normal table (Z-table) provides the probability that a standard normal variable is less than a given value.

• Finding Percentiles: To find the value x corresponding to a percentile p:

$x = \mu + z_p \sigma$

• where $z_p$ is the z-score for percentile p.

• Example: For μ = 100, σ = 15, the 90th percentile is $x = 100 + (1.28)(15) = 119.2$ (using $z_{0.90} \approx 1.28$).

Chapter 8.1: Sampling Distribution of the Sample Mean

Sampling Distribution

The sampling distribution of the sample mean describes the distribution of sample means from all possible samples of a fixed size n drawn from a population.

• Mean of Sampling Distribution: $\mu_{\bar{x}} = \mu$

• Standard Deviation (Standard Error):

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

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

• Calculating Probabilities: Use the sampling distribution to find probabilities about $\bar{x}$.

• Example: If μ = 50, σ = 10, n = 25, then $\mu_{\bar{x}} = 50$, $\sigma_{\bar{x}} = 10/5 = 2$.

Chapter 9.2: Confidence Intervals

Constructing and Interpreting Confidence Intervals

A confidence interval estimates a population parameter (such as the mean) with an associated level of confidence.

• Purpose: To provide a range of plausible values for the population parameter.

• Interpretation: A 95% confidence interval means that, in repeated sampling, 95% of such intervals will contain the true parameter.

• Formula (t-distribution):

$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

• where $t^*$ is the critical value from the t-distribution with n-1 degrees of freedom, s is the sample standard deviation.

• Determining Sample Size: To achieve a specified margin of error (E):

$n = \left( \frac{z^* \sigma}{E} \right)^2$

• Example: For $\bar{x} = 20$, $s = 4$, $n = 16$, 95% confidence interval is $20 \pm t^* \frac{4}{4}$.

Chapter 10.1 and 10.3: Hypothesis Testing

General Concepts

Hypothesis testing is a statistical method for making decisions about population parameters based on sample data.

• Null Hypothesis (H0): The default assumption (e.g., μ = μ0).

• Alternative Hypothesis (Ha): The competing claim (e.g., μ ≠ μ0).

• Significance Level (α): The probability of rejecting H0 when it is true (commonly 0.05).

• Test Statistic: Measures how far the sample statistic is from the null hypothesis value, in standard errors.

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

• Critical Region: The set of values for which H0 is rejected.

• Classical Approach: Compare test statistic to critical value(s).

• p-value Approach: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under H0.

• Conclusion: If p-value < α, reject H0; otherwise, fail to reject H0.

• Example: Testing if μ = 100, sample mean = 104, s = 8, n = 16, α = 0.05. Compute t, compare to critical value or find p-value.

Type I and Type II Errors, Power

• Type I Error (α): Rejecting H0 when it is true.

• Type II Error (β): Failing to reject H0 when Ha is true.

• Power: Probability of correctly rejecting H0 (1 - β).

• Relationships: Lowering α increases β (and vice versa); increasing sample size increases power.

• Correct Decision: When the test conclusion matches the true state of nature.

Chapter 11.2/11.3: Two-Sample Hypothesis Tests for Population Means

Comparing Two Population Means

Two-sample tests compare the means of two populations, using either independent or dependent samples.

• Estimating Difference: Estimate μ1 - μ2 using sample means.

• Independent Samples: Samples from two unrelated groups.

• Dependent Samples (Paired): Each observation in one sample is paired with an observation in the other (e.g., before/after studies).

Paired t-Test (Dependent Samples)

• Test Statistic:

$t = \frac{\bar{d}}{s_d/\sqrt{n}}$

• where $\bar{d}$ is the mean of the differences, $s_d$ is the standard deviation of the differences, n is the number of pairs.

Pooled t-Test (Independent Samples, Equal Variances)

• Pooled Standard Deviation:

$s_p = \sqrt{\frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}}$

• Test Statistic:

$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

Welch's t-Test (Independent Samples, Unequal Variances)

• Test Statistic:

$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

• Degrees of Freedom: Calculated using the Welch-Satterthwaite equation.

$df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1-1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2-1}}$

• Example: Comparing means of two independent groups with different variances.

Summary Table: Types of Two-Sample t-Tests

Test	Sample Type	Variance Assumption	Test Statistic Formula
Paired t-Test	Dependent	Not applicable	$t = \frac{\bar{d}}{s_d/\sqrt{n}}$
Pooled t-Test	Independent	Equal variances	$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$
Welch's t-Test	Independent	Unequal variances	$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

Additional info: This guide expands on the exam topic list by providing definitions, formulas, and examples for each concept. For full mastery, students should practice applying these methods to sample problems and interpreting results in context.