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7: The Central Limit Theorem

Last updated

Mar 17, 2025
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- 6.11: Solutions
- 7.0: Introduction to the Central Limit Theorem

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7.0: Introduction to the Central Limit Theorem
Why are we so concerned with means? Two reasons are: they give us a middle ground for comparison, and they are easy to calculate. In this chapter, you will study means and the Central Limit Theorem.
7.1: The Central Limit Theorem for Sample Means
The Central Limit Theorem answers the question: from what distribution did a sample mean come? If this is discovered, then we can treat a sample mean just like any other observation and calculate probabilities about what values it might take on. We have effectively moved from the world of statistics where we know only what we have from the sample, to the world of probability where we know the distribution from which the sample mean came and the parameters of that distribution.
7.2: Using the Central Limit Theorem
This page explains the Central Limit Theorem (CLT), which asserts that larger sample sizes lead to a normal sampling distribution of the sample mean, regardless of the population's original distribution. It highlights the Law of Large Numbers, which states that larger samples yield means closer to the population mean.
7.3: The Central Limit Theorem for Proportions
This page explains the Central Limit Theorem, describing how sample means and proportions derive from normal distributions. The sample proportion $\hat{p}$ is generated from binomial data, leading to its own theoretical distribution with a mean equal to the population proportion $p$ and a standard deviation $\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$ . It highlights that increasing sample size $n$ reduces standard deviation, resulting in more precise estimates.
7.4: Finite Population Correction Factor
This page discusses how sample size affects the variance and standard deviation of sampling distributions, stressing the Finite Population Correction Factor when sampling over 5% of a population. It emphasizes this adjustment's significance for means and proportions, providing two examples: weights of German Shepherds and customers exceeding credit limits. These examples illustrate using the correction factor to calculate probabilities for sample means and proportions.
7.5: Limit Theorem (Worksheet)
A statistics Worksheet: The student will demonstrate and compare properties of the central limit theorem.
7.6: Key Terms
This page defines essential statistical terms, such as average, normal distribution, and central limit theorem. It explains averages as central tendency measures and how sample means approximate normal distribution with larger sample sizes. Key concepts like standard error associated with sample means and proportions are also introduced.
7.7: Chapter Review
This page discusses the Central Limit Theorem (CLT), which asserts that as sample sizes increase, the distribution of sample means approximates a normal distribution, with the sample mean aligning with the population mean. The standard error of the mean is determined using the population standard deviation and sample size. CLT reinforces the law of large numbers, showing that larger samples produce means that are more accurate representations of the population mean.
7.8: Formula Review
This page explains the Central Limit Theorem for Sample Means, which states that as the sample size increases, the distribution of sample means approaches a normal distribution, characterized by mean $\mu_{\overline{x}}$ and standard error $\frac{\sigma}{\sqrt{n}}$ .
7.9: Practice
This page presents exercises focused on the Central Limit Theorem, exploring weights and battery life distributions. It covers uniform and exponential distributions, expected values, standard deviations, and percentiles. Examples include calculating probabilities and applying the finite population correction factor. The exercises encourage the use of statistical concepts to analyze sampling distributions with known population parameters.
7.10: Homework
This page explores the Central Limit Theorem (CLT) and its significance in analyzing sample means from several distributions, emphasizing the role of sample size in determining mean, standard deviation, and distribution of averages. It includes practical examples from diverse scenarios like student change amounts, survey results, and marathon times.
7.11: References
7.12: Solutions
This page outlines statistical analyses, featuring measures like mean and standard deviation across scenarios. It discusses probability distributions, including normal and exponential types, and highlights sampling distribution properties, particularly the Central Limit Theorem. The content focuses on calculations related to statistical concepts and hypothesis testing, emphasizing various probabilities tied to these distributions.

Curated and edited by Kristin Kuter | Saint Mary's College, Notre Dame, IN

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