8: Hypothesis Testing for One Sample
- Page ID
- 46106
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 8.1: Introduction to Hypothesis Testing
- Hypothesis testing is a method used to make decisions or draw conclusions about a population based on sample data. The process involves stating hypotheses, choosing a significance level, selecting a test, finding the test value, and making a decision. Tests can be left-tailed (less than), right-tailed (greater than), or two-tailed (not equal).
- 8.2: Type I and II Errors
- In hypothesis testing, a Type I error occurs when a true null hypothesis is wrongly rejected, while a Type II error happens when a false null hypothesis is not rejected. Type I is a false positive; Type II is a false negative. The risk of these errors is managed through significance levels and sample design.
- 8.3: z-Test for a Mean
- A z-test is used in hypothesis testing when the population standard deviation is known and the sample size is large or the population is normal. The critical value method compares the test statistic to cutoff values to make a decision. The p-value method compares the probability of the result to the significance level.
- 8.4: t-Test for a Mean
- A t-test is used for hypothesis testing when the population standard deviation is unknown and the sample size is small. It relies on the t-distribution, which adjusts for extra variability. The critical value method involves comparing the test statistic to a t-value from the distribution to decide whether to reject the null hypothesis.
- 8.5: z-Test for a Proportion
- A z-test for proportions is used when testing claims about population proportions with a large enough sample size to assume normality. It compares the sample proportion to the claimed proportion. The critical value method involves comparing the test statistic to a z-value cutoff to decide whether to reject the null hypothesis.
- 8.6: Formulas for Chapter 8
- This section lists the key formulas for Chapter 8 and explains their uses. These formulas are essential for conducting hypothesis tests involving population means and proportions. The z-test for a mean is used when the population standard deviation is known, while the t-test is used when it is unknown. For categorical data, the z-test for a proportion determines whether a sample proportion significantly differs from a claimed population value.
- 8.7: Chapter 8 Key Terms and Symbols
- This section presents the key terms and symbols for Chapter 8, which covers one-sample hypothesis testing for means and proportions. It includes essential terminology related to statistical hypotheses, error types, significance levels, test statistics, and sampling distributions. These elements provide the foundation for understanding and applying hypothesis testing methods.