9: Inferences with Two Samples
- Page ID
- 10982
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)You have learned to conduct hypothesis tests on single means and single proportions. You will expand upon that in this chapter. You will compare two means or two proportions to each other. The general procedure is still the same, just expanded. To compare two means or two proportions, you work with two groups. The groups are classified either as independent or matched pairs. Independent groups consist of two samples that are independent, that is, sample values selected from one population are not related in any way to sample values selected from the other population. Matched pairs consist of two samples that are dependent. The parameter tested using matched pairs is the population mean. The parameters tested using independent groups are either population means or population proportions.
- 9.1: Prelude to Hypothesis Testing with Two Samples
- This page covers hypothesis testing for two groups, teaching students to classify and analyze independent and matched samples. Key topics include testing two means and two proportions, as well as handling paired samples. The differences between independent and matched pairs are highlighted, alongside the importance of using calculators or software for calculations, thereby enhancing students' understanding of hypothesis tests related to single means and proportions.
- 9.2: Inferences for Two Population Means- Large, Independent Samples
- This page explains estimation and hypothesis testing for the difference between the means of two populations, emphasizing the use of large independent samples. It details how to construct confidence intervals and perform hypothesis tests, providing a real-world example of customer satisfaction comparisons. Additionally, it covers the computation of the test statistic (Z), rejection regions, and p-value approaches for hypothesis testing, confirming decisions through examples.
- 9.3: Inferences for Two Population Means - Unknown Standard Deviations
- This page explains constructing confidence intervals and conducting hypothesis tests for the difference between means of two independent populations with unknown standard deviations, highlighting the t-distribution's role. It provides formulas for calculating test statistics and degrees of freedom, especially with unequal variances, and illustrates the methods with examples.
- 9.4: Inferences for Two Population Means - Paired Samples
- This page covers statistical analysis of independent versus paired samples, emphasizing confidence intervals and hypothesis testing for mean differences. It uses fuel economy comparisons as a practical example to demonstrate paired sampling, which reduces variability. The page details formulating null and alternative hypotheses, calculating test statistics via Student's t-distribution, and interpreting results.
- 9.5: Inferences for Two Population Proportions
- This page covers the construction of confidence intervals and hypothesis testing for comparing proportions of two populations. It emphasizes the necessity for random samples and presents a formula for the confidence interval, requiring large sample sizes. An example illustrates passing rates before and after public access to inspection records.
- 9.6: Which Analysis Should You Conduct?
- This page highlights the significance of choosing the correct statistical analysis tailored to specific questions, emphasizing factors like mean versus proportion, sample count, and sample independence. It distinguishes between hypothesis tests and confidence intervals based on problem language, underscoring the necessity of grasping these differences for proper statistical testing and appropriate application of assumptions and calculations.
- 9.E: Hypothesis Testing with Two Samples (Optional Exercises)
- These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax.
Contributors and Attributions
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/30189442-699...b91b9de@18.114.