7: Sampling, Standard Error, Central Limit Theorem
- Page ID
- 50634
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By the end of this chapter, you will be able to:
- Explain why we need samples to represent the population
- Apply the guidelines for sample sizes
- Examine the consequences of sampling bias
- Explain the need for the standard error of the mean
- Explain the purpose of the Central Limit Theorem (CLT)
Key Terms:
- Sample and the Population
- Sampling
- Sample Bias
- Standard Error of the Mean
- Central Limit Theorem
- 7.1: The Purpose of Sampling
- This page emphasizes the significance of estimating population parameters in statistics, focusing on how sample statistics inform about larger populations. It addresses challenges in ensuring samples represent populations, such as sample size and measurement issues in psychology. Establishing a baseline is vital for identifying significant observations. However, logistical and methodological obstacles hinder accurate assessments, especially in sensitive contexts.
- 7.2: Sampling from the Population
- This page emphasizes the importance of large sample sizes in research to accurately represent populations and ensure reliable predictions. It stresses the limitations of small samples, particularly a sample size of one, in drawing conclusions. The text highlights the need for diverse sampling locations and contexts, illustrated through examples like temperature readings in Chicago and the challenges in estimating populations with eating disorders.
- 7.3: Getting a Good Sample - Get a Random Sample
- This page emphasizes the significance of obtaining a representative sample in research through effective design and strategies. It addresses challenges in recruiting specific populations like police officers with PTSD and LGBTQ+ individuals, often affected by stigma. The text advocates for random sampling as the best approach to ensure unbiased representation, criticizing convenience sampling for its potential bias.
- 7.4: Sampling Bias or Error
- This page discusses sampling bias, which happens when a sample misrepresents the population, resulting in flawed conclusions. It notes that this bias is inevitable due to the absence of a definitive population mean for comparison. Increasing sample size can improve representation, as larger samples yield a sampling distribution closer to the true population distribution.
- 7.5: Central Limit Theorem (CLT) and Its Implications
- This page explains the Central Limit Theorem (CLT), which states that larger sample sizes lead to sample means approximating a normal distribution, enabling accurate representation of the population. As sample sizes increase, the standard error decreases, improving estimations of the population mean.
- 7.6: Summary of Reasons Why You Need a Large Sample Size
- This page emphasizes the importance of large sample sizes for accurate statistical conclusions, highlighting risks of Type I and II errors with small samples. It refers to the Central Limit Theorem and standard error of the mean to support the need for larger samples. Recommended guidelines suggest a minimum of 30 participants and a maximum of 300, with five per survey item.
- 7.7: Discussion Questions
- This page explains the distinction between standard error and standard deviation, highlighting that the former assesses sample mean accuracy in estimating a population mean, while the latter indicates data variability. It emphasizes the importance of sample size, noting that small samples can lead to unreliable, skewed results. Larger sample sizes enhance accuracy and reduce variability, thereby boosting statistical reliability and generalizability.