10: Hypothesis Testing with Two Samples
- Page ID
- 45728
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 10.0: Introduction to Two-Sample Tests
- This page explains hypothesis testing in comparing two groups, like in health or education studies. It distinguishes between independent groups (unrelated samples) and matched pairs (dependent samples), noting that matched pairs assess population means while independent groups can evaluate means or proportions. The methodology is applicable across diverse fields such as health, politics, and business.
- 10.1: Comparing Two Independent Population Means
- This page discusses testing hypotheses for mean differences between independent populations, focusing on the Aspin-Welch \(t\)-test for unequal variances. It highlights the Central Limit Theorem's role in sample means distribution, illustrating standard error and test statistics calculation through a case study on coconut processing shifts.
- 10.2: Cohen's Standards for Small, Medium, and Large Effect Sizes
- This page discusses Cohen's \(d\), a statistic for measuring effect size by comparing the difference between two means relative to their pooled standard deviation. It categorizes effects as small (0.2), medium (0.5), or large (0.8) but does not provide confidence intervals or significance levels. An example illustrates a \(d\) value of 0.384, indicating a small effect and minimal differences between the two groups' means.
- 10.3: Test for Differences in Means- Assuming Equal Population Variances
- This page explores hypothesis testing for mean differences with unknown population variances, highlighting the use of pooled sample variances for accuracy. An example illustrates testing a drug's effectiveness in boosting endorphin production against a placebo. The null hypothesis posits no mean difference, while the alternative asserts the drug is more effective.
- 10.4: Comparing Two Independent Population Proportions
- This page discusses the comparison of two independent population proportions, emphasizing the need for random, independent samples with adequate successes and failures. It explains the use of hypothesis testing to examine true differences in proportions, with the null hypothesis often testing for equality. The pooled proportion aids in calculating the test statistic.
- 10.5: Two Population Means with Known Standard Deviations
- This page covers hypothesis testing for independent means with known population standard deviations, using examples of floor wax efficacy and U.S. senator ages. It establishes null hypotheses to compare means and finds insufficient evidence to support alternative hypotheses, concluding that wax 1 does not outperform wax 2 and that Democratic senators are not older than their Republican counterparts at a 5% significance level.
- 10.6: Matched or Paired Samples
- This page explores business data analysis through matched pair hypothesis tests, illustrating their use in evaluating interventions like training programs and treatments. It highlights the application of statistical tests, particularly the Student's-t test, in two scenarios: the successful assessment of hypnotism's effectiveness on sensory measurements and the lack of evidence for performance improvement in college football players after a strength development class.
- 10.7: Key Terms
- This page defines key statistical terms: Cohen's d measures effect size (small, medium, large); independent groups involve two unrelated samples, while matched pairs consist of dependent samples for comparison; pooled variance is a weighted average of two variances, essential for calculating standard error.
- 10.8: Chapter Review
- This page discusses methods for comparing two independent population means and proportions, addressing both known and unknown population standard deviations. It introduces Cohen's \(d\) as an effect size measure and highlights the importance of equal variance assumptions. The text specifies distribution characteristics for various statistical tests, such as the Student's \(t\)-distribution and normal distribution based on data conditions.
- 10.9: Formula Review
- This page discusses statistical methods for comparing two independent population means, focusing on standard error, test statistics, and effect sizes. It includes formulas for calculating standard error and \(t\)-scores, determining degrees of freedom, and introduces Cohen's \(d\) for effect size. The text covers cases with known variances and matched samples, detailing the use of \(z\)-scores and \(t\)-scores based on variance knowledge.
- 10.10: Practice
- This page presents hypothesis testing exercises covering scenarios with independent population means and proportions, including examples from drivers' test pass rates, consumer preferences, salary differences, and life expectancy. It also addresses statistical tests related to baseball pitching speeds, plant heights, and medication effectiveness.
- 10.11: Homework
- This page outlines various statistical experiments and hypothesis tests comparing means and proportions across different contexts, including studies on college course enrollments, breath-holding times, salary comparisons, smartphone usage, and the effectiveness of diets and interventions. It emphasizes the use of statistical methods to assess the significance of observed differences in multiple scenarios, such as marital satisfaction, sports outcomes, and health-related studies.
- 10.12: Bringing It Together Homework
- This page provides an overview of hypothesis tests for different research scenarios, including independent group means, paired samples, single means, and proportions. It describes various contexts like dietary studies and course performance analysis while outlining the classification criteria for the tests, emphasizing the nature of the samples involved.
- 10.13: References
- This page details various data sources for comparing independent population means and proportions, mentioning links from organizations including Graduating Engineer, Microsoft Bookshelf, the US Senate, Nasdaq, and health entities like the CDC and American Cancer Society. It also references topics such as crime rates, smartphone use, obesity prevalence, and gender differences in sexting, all of which were accessed in June 2013.
- 10.14: Solutions
- This page covers statistical hypothesis tests examining proportions and means across various scenarios, focusing on the evaluation of null (H0) versus alternative hypotheses (Ha). It includes analyses of auto insurance costs, vehicle fuel efficiency, couple satisfaction, breast cancer incidence, and underemployment rates.
Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN