6: Inferential Statistics and an Introduction to Hypothesis Testing
- Page ID
- 287712
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 6.1: Growth Mindset
- What's growth mindset?
- 6.2: People, Samples, and Populations
- This page covers individual scores, samples, and their connection to population parameters in statistics. It explains \(z\)-scores for locating individual scores in distributions and introduces variance and standard deviation as measures of variability.
- 6.3: The Sampling Distribution of Sample Means
- This page covers sampling distributions, focusing on sample means from repeated samples. It discusses key characteristics like normal shape, true population mean, and standard error. The central limit theorem and law of large numbers are highlighted as essential principles.
- 6.4: Using Standard Error for Probability
- This page explores the use of \(z\)-scores in assessing sample means from a normal distribution, focusing on standard error's role, and demonstrates that larger sample sizes decrease the probability of extreme sample means.
- 6.5: Sampling Distribution, Probability and Inference
- This page covers the standard error concept in probability analysis, emphasizing its importance in reflecting variations in statistics like the sample mean versus the population mean.
- 6.6: Logic and Purpose of Hypothesis Testing
- This page covers hypothesis testing through two examples that indicate that observed effects are likely not due to chance.
- 6.7: The Probability Value
- This page emphasizes the importance of understanding probability values in statistics.
- 6.8: Descriptive versus Inferential Statistics
- What are inferential statistics, and how do they differ from what we've been doing?
- 6.9: The Null Hypothesis
- This page explains the null hypothesis, which posits that observed effects are due to chance, asserting no difference in population parameters like means or correlations.
- 6.10: The Alternative Hypothesis
- This page explains the alternative hypothesis that is adopted upon rejecting the null hypothesis. It differentiates between directional alternatives and non-directional alternatives.
- 6.11: Null Hypothesis Significance Testing
- What do we do with the Research Hypothesis and the Null Hypothesis?
- 6.12: Critical Values, p-values, and Significance
- Let's start putting it altogether so that we can start answering research questions.
- 6.13: Steps of the Hypothesis Testing Process
- Four easy steps!
- 6.14: Movie Popcorn
- An example.
- 6.15: Effect Size
- This page emphasizes the significance of effect sizes in statistical analysis, especially after null hypothesis rejection. It highlights that while statistical significance shows a difference, it does not imply practical relevance. Effect sizes, like Cohen’s \(d\), measure the effect's magnitude and categorize it as small, moderate, or large, enhancing the interpretation of results.
- 6.16: Office Temperature
- An example.
- 6.17: The Two Errors in Null Hypothesis Significance Testing
- When working with probabilities, there's always a chance that we'll make the wrong decision and make a mistake (in other words, an error)...