Unit 6 Review: Inference for Categorical Data and Proportions

Significance Test

A formal procedure for comparing observed data with a claim (hypothesis) to assess its truth.

Null Hypothesis ($H_0$)

The claim of 'no difference' or 'no effect' that includes an equality sign.

Alternative Hypothesis ($H_a$)

The claim we hope is true, looking for evidence against the null.

P-value

The probability of obtaining a test statistic as extreme as the observed statistic, assuming $H_0$ is true.

Significance Level ($\alpha$)

The threshold for rejecting the null hypothesis, commonly set at 0.05, 0.01, or 0.10.

Type I Error

Rejecting $H_0$ when it is actually true.

Type II Error

Failing to reject $H_0$ when it is actually false.

Power of a Test

The probability that the test correctly rejects a false null hypothesis.

One-Sample $z$-Test for Proportion

A test used for claims about a single proportion.

Critical Value

The value that the test statistic must exceed to reject the null hypothesis.

Large Counts Condition

Requires $n p<em>0 \geq 10$ and $n (1 - p</em>0) \geq 10$ for roughly Normal distribution.

Pooled Proportion ($\hat{p}_c$)

The combined estimate of proportions used in two-sample tests when $H_0$ is true.

Test Statistic ($z$)

Measures how many standard deviations the sample proportion is from the null parameter.

Decision Rule for Tests

If $P$-value \leq $\alpha$, reject $H<em>0$; if $P$-value > $\alpha$, fail to reject $H</em>0$.

Failure to Reject $H_0$

Insufficient evidence to claim an effect or difference.

Random Condition

The data must come from a random sample or randomized experiment.

10% Condition

Ensures independence when sampling without replacement: $n \leq \frac{1}{10}N$.

Effect Size

The magnitude of difference that indicates a meaningful effect in hypothesis testing.

Standard Normal Table

A table used to find probabilities associated with calculated $z$-scores.

Evidence Against $H_0$

Small P-values indicate strong evidence against the null hypothesis.

Restating $H_a$ in Context

Rewording the alternative hypothesis to fit the specifics of the problem being analyzed.

Conclusion of a Test

A summary statement reflecting the comparison of P-value and $\alpha$.

Hypothesis Testing Process

The logical framework following 'State, Plan, Do, Conclude' to test claims.

Independent Samples

Samples that are not influenced by each other, necessary for valid two-sample tests.

Equivalence of Confidence Intervals and Tests

If a CI for $p<em>1 - p</em>2$ does not include 0, it is equivalent to rejecting $H_0$.

Common Statistical Mistakes

Errors often made in hypothesis notation, accepting null, and using improper parameters.