Comprehensive Study Notes for Hypothesis Testing, Confidence Intervals, and Probability in Statistics

Hypothesis Testing

Types of Errors in Hypothesis Testing

In hypothesis testing, two types of errors can occur when making decisions about the null hypothesis (H0):

• Type I Error (α): Rejecting the null hypothesis when it is actually true.

• Type II Error (β): Failing to reject the null hypothesis when it is actually false.

Example: If a test incorrectly concludes that a new drug is effective when it is not, this is a Type I error.

Formulating Hypotheses

Hypothesis testing involves two competing statements:

• Null Hypothesis (H0): The statement being tested, usually a statement of no effect or status quo.

• Alternative Hypothesis (Ha): The statement we want to find evidence for.

Example: Testing if the mean weight is greater than 50 kg:

• H0: μ = 50

• Ha: μ > 50

Decisions in Hypothesis Testing

There are two possible decisions:

• Reject H0: Sufficient evidence exists to support Ha.

• Fail to Reject H0: Insufficient evidence to support Ha.

Significance Level (α)

The significance level (α) is the probability of making a Type I error. Common values are 0.05 or 0.01. Lowering α decreases the probability of a Type I error but increases the probability of a Type II error.

P-Value Interpretation

The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value, assuming H0 is true. If the p-value is less than α, reject H0.

Critical Value Approach

Compare the test statistic to a critical value from the appropriate distribution (z or t). If the test statistic falls in the rejection region, reject H0.

Test Statistic Formulas

• z-test (known σ): $z = \frac{\bar{x} - \mu_0}{\sigma/\sqrt{n}}$

• t-test (unknown σ): $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

Example Table: Hypothesis Test Summary

Step	Description
State Hypotheses	H0: μ = μ0; Ha: μ ≠ μ0
Assumptions	Population is normal or n > 30
Find Critical Value	Use z or t table for α
Calculate Test Statistic	Use formula above
Decision	Compare test statistic to critical value
Conclusion	State in context of problem

Confidence Intervals

Definition and Interpretation

A confidence interval provides a range of plausible values for a population parameter. A 95% confidence interval means that, in repeated sampling, 95% of such intervals will contain the true parameter.

Formulas

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

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

Margin of Error (E)

The margin of error is the maximum expected difference between the true population parameter and a sample estimate.

• $E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$ (for known σ)

• $E = t_{\alpha/2, df} \frac{s}{\sqrt{n}}$ (for unknown σ)

Sample Size Determination

To achieve a desired margin of error, solve for n:

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

Example: To estimate a mean with σ = 10, E = 2, and 95% confidence (z = 1.96):

• $n = \left( \frac{1.96 \times 10}{2} \right)^2 = 96.04$ (round up to 97)

Probability Rules

Multiplication Rule

• Independent Events: $P(A \text{ and } B) = P(A) \times P(B)$

• Dependent Events: $P(A \text{ and } B) = P(A) \times P(B|A)$

Addition Rule

• Mutually Exclusive: $P(A \text{ or } B) = P(A) + P(B)$

• Not Mutually Exclusive: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$

Discrete Probability Distributions

Binomial Distribution

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

Where n = number of trials, p = probability of success, q = 1 - p.

Normal Distribution and Standard Scores

Standard Normal Distribution

The standard normal distribution is a normal distribution with mean 0 and standard deviation 1. Z-scores measure how many standard deviations a value is from the mean:

• $z = \frac{x - \mu}{\sigma}$

Use the Standard Normal Table to find probabilities associated with z-scores.

Student's t-Distribution

Used when the population standard deviation is unknown and the sample size is small (n < 30). The t-distribution is wider than the normal distribution and depends on degrees of freedom (df = n - 1).

Correlation and Regression

Correlation Coefficient (r)

Measures the strength and direction of a linear relationship between two variables:

• $r = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$

Regression Line

The regression line predicts the value of a dependent variable based on the value of an independent variable:

• $y = mx + b$

• $m = \frac{n\sum xy - \sum x \sum y}{n\sum x^2 - (\sum x)^2}$

• $b = \frac{\sum y - m\sum x}{n}$

Statistical Tables

Standard Normal Table

Provides cumulative probabilities for z-scores. Used to find probabilities and critical values for hypothesis testing and confidence intervals.

t-Distribution Table

Gives critical values for the t-distribution based on degrees of freedom and significance level.

Summary Table: Key Formulas

Concept	Formula
z-score	$z = \frac{x - \mu}{\sigma}$
t-score	$t = \frac{\bar{x} - \mu}{s/\sqrt{n}}$
Confidence Interval (mean, known σ)	$\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
Confidence Interval (mean, unknown σ)	$\bar{x} \pm t_{\alpha/2, df} \frac{s}{\sqrt{n}}$
Sample Size (mean, known σ)	$n = \left( \frac{z_{\alpha/2} \sigma}{E} \right)^2$
Binomial Mean	$\mu = np$
Binomial Variance	$\sigma^2 = npq$

Applications and Examples

• Calculating minimum sample size for a desired margin of error.

• Constructing and interpreting confidence intervals for means.

• Performing hypothesis tests for population means using z or t statistics.

• Using statistical tables to find critical values and probabilities.

Additional info: These notes synthesize content from a practice final exam, formula sheets, and statistical tables, covering hypothesis testing, confidence intervals, probability rules, and applications relevant to a college-level statistics course.