8.7: Vocabulary (Chapter 8)
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 8 Vocabulary
- Hypothesis
- A testable statement about a population parameter. Hypotheses are used to structure statistical tests and are evaluated based on sample data.
- Null Hypothesis (\( H_0 \))
- The default or starting assumption in a hypothesis test — usually a statement of “no effect,” “no change,” or “no difference.” We test this claim using data.
- Alternative Hypothesis (\( H_A \))
- The rival claim to the null. It represents the outcome we're trying to find evidence for (e.g., an increase, decrease, or difference).
- Test Statistic
- A calculated value (e.g., z, t) that measures how far the sample statistic is from the null hypothesis value. It's used to determine how extreme the result is.
- z-test for a mean
- A hypothesis test used when the population standard deviation is known. Assumes normal distribution. The test statistic is:
\( Z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}} \) - Welch’s t-test
- A two-sample test for comparing means when variances are unequal and sample sizes may differ. It uses sample standard deviations and calculates degrees of freedom based on variability.
- z-test for a proportion
- Used to test a hypothesis about a population proportion. The test statistic is:
\( Z = \frac{\hat{p} - p_0}{\sqrt{p_0(1 - p_0)/n}} \) - p-value
- The probability of observing a result as extreme or more extreme than your observed sample statistic, assuming the null hypothesis is true. A small p-value suggests the data are inconsistent with \( H_0 \).
- Significance Level (\( \alpha \))
- The threshold we set for how much risk of a Type I error we are willing to accept. Common values are 0.05, 0.01, and 0.10.
- Statistical Significance
- If the p-value is less than or equal to \( \alpha \), we say the result is statistically significant — meaning it is unlikely to have occurred by chance under the null.
- Practical Significance
- A result that is meaningful or impactful in the real world. Even small statistical differences may not matter practically, depending on the context.
- Type I Error
- Rejecting the null hypothesis when it is actually true. This is a false positive. The probability of this is controlled by \( \alpha \).
- Type II Error
- Failing to reject the null hypothesis when the alternative is actually true. This is a false negative.
- Power of a Test
- The probability that a test will correctly reject a false null hypothesis. It is equal to 1 minus the probability of a Type II Error.
- One-tailed Test
- A hypothesis test in which the alternative hypothesis is directional (e.g., \( p > p_0 \) or \( \mu < \mu_0 \)) — we are only interested in one direction of difference.
- Two-tailed Test
- A hypothesis test in which the alternative hypothesis tests for any difference from the null (e.g., \( \mu \ne \mu_0 \)). Both tails of the distribution are considered.
- Decision Rule
- The rule that guides whether to reject or fail to reject the null hypothesis, usually based on comparing the p-value to \( \alpha \).
- p-hacking
- A problematic practice in which researchers run many statistical analyses until they obtain a significant (p < 0.05) result, often misrepresenting the evidence.
- Effect Size
- A measure of how big a difference or change is — separate from statistical significance. A small p-value may accompany a small effect size, or vice versa.
- Standard Error
- The estimated standard deviation of a sampling distribution. Used in many test statistics, including for means and proportions.