8: Confidence Intervals
- Page ID
- 4595
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 8.0: Introduction to Confidence Intervals
- This page explains inferential statistics, emphasizing point estimates and confidence intervals, with an example of estimating the mean iTunes downloads. It discusses the Central Limit Theorem and the Empirical Rule's roles in confidence intervals and presents the formula for calculating them for population means.
- 8.1: A Confidence Interval for a Population Standard Deviation Known or Large Sample Size
- A confidence interval for a population mean with a known population standard deviation is based on the conclusion of the Central Limit Theorem that the sampling distribution of the sample means follow an approximately normal distribution.
- 8.2: A Confidence Interval for a Population Standard Deviation Unknown, Small Sample Case
- This page discusses the challenges of estimating population parameters without a known standard deviation, especially with small sample sizes. It details the development of the Student's t-distribution for improved confidence interval accuracy, emphasizing its resemblance to the normal distribution as sample sizes increase.
- 8.3: A Confidence Interval for A Population Proportion
- This page covers confidence intervals for population proportions, particularly in election polling and market research. It details the calculation process using estimated sample proportions, standard deviation, and confidence levels, emphasizing binomial distributions and the Central Limit Theorem. Two examples illustrate confidence interval calculations: one estimating the proportion of dogs in professional events and another for overdue accounts.
- 8.4: Calculating the Sample Size n- Continuous and Binary Random Variables
- This page covers sample size determination for estimating population parameters with continuous and binary random variables. It provides formulas based on confidence levels and acceptable error, along with cautionary notes for binary variables. A table displays required sample sizes for different confidence levels and tolerances, while an example showcases sample size calculation for a mobile phone company's survey.
- 8.5: Confidence Intervals - Inflation Rates (Worksheet)
- A statistics Worksheet: The student will calculate a 90% confidence interval using the given data. The student will determine the relationship between the confidence level and the percentage of constructed intervals that contain the population mean.
- 8.6: Key Terms
- This page defines key statistical concepts like binomial distribution, confidence intervals, and normal distribution. It explains their characteristics and implications for statistical analysis, focusing on the relationship between sample statistics and population parameters, as well as the behavior of distributions concerning sample size.
- 8.7: Chapter Review
- This page outlines how to create confidence intervals when the population standard deviation is unknown, focusing on small samples. It discusses the use of Student’s t-distribution, provides the formula for calculating t-scores and confidence intervals, and explains the method for estimating population proportions. It also details the process to calculate sample size needed for estimating a population mean or proportion within a specified margin of error.
- 8.8: Formula Review
- This page discusses methods for calculating confidence intervals for population parameters, focusing on small samples. It emphasizes using the t-score and Student's t-distribution for unknown population standard deviations, as well as sample proportions for estimating population proportions. The text includes formulas for calculating confidence intervals and determining necessary sample sizes for both continuous and binary data.
- 8.9: Practice
- This page outlines exercises on confidence intervals for population means and proportions, detailing calculations for sample statistics, confidence levels, and error bounds. It emphasizes the impact of sample size and confidence levels on these intervals, while also addressing challenges in surveys and the need for proper identification of random variables and distributions.
- 8.10: Homework
- This page discusses the construction and interpretation of confidence intervals across various statistical problems, including proportions, means, and surveys. It emphasizes the impact of sample size, confidence levels, and margin of error on accuracy. Exercises cover defining random variables, calculating sample statistics, and interpreting results in contexts such as college enrollments and campaign contributions.
- 8.11: References
- This page provides a compilation of sources for confidence intervals associated with population standard deviations and proportions. It categorizes these sources based on sample size and includes links to reputable entities like the U.S. Census Bureau and the CDC, covering various topics such as demographics, health trends, and economic surveys, with all data accessed in July 2013.
- 8.12: Solutions
- This page covers statistical concepts such as confidence intervals, sample means, and error bounds. It explores mean wait times in emergency rooms, television habits, executive preferences, and the impact of sample sizes on population estimates. The text highlights the use of Student’s t-distribution for small samples and variability implications.
Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN