Sampling Distribution Overview

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1. Sampling Distribution
   1. Introduction
   2. Sampling Distribution for Arithmetic Mean
   3. Sampling Distribution for Proportion of Success </aside>

Sampling Distribution

1. Sampling Distribution

   1. Introduction
      1. Population: The set of all elements of interest in a study.
      2. Sample: A subset of the population used for data collection and analysis.
      3. Sample Statistic: A measure (e.g., mean, standard deviation) computed from a sample.
      4. Sampling Distribution: The probability distribution of a sample statistic (e.g., sample mean, sample proportion).
   2. Key Idea
      1. Different samples drawn from the same population may provide different values for the sample statistic.
      2. The collection of all possible values of the sample statistic forms the sampling distribution.

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2. Sampling Distribution of Arithmetic Mean

   1. Definition
      1. Sampling Distribution of Arithmetic Mean: The probability distribution of all possible values of the sample mean $\bar{x}$ when samples of size $n$ are drawn from a population.
   2. Mean of Sampling Distribution (Expected Value)
      1. $E(\bar{x}) = \mu$,
      2. where:
         1. $E(\bar{x})$: Expected value of the sample mean.
         2. $\mu$: Population mean.
      3. If the expected value of the sample mean equals the population mean, the sample mean is an unbiased estimator of $\mu$.

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3. Standard Deviation of Sampling Distribution (Standard Error of the Mean)

   1. For a finite population:

      $$ \sigma_{\bar{x}} = \sqrt{\frac{N - n}{N - 1}} \cdot \frac{\sigma}{\sqrt{n}} $$

      1. Where:
         1. $\sigma_{\bar{x}}$: Standard error of the mean.
         2. $N$: Population size.
         3. $n$: Sample size.
         4. $\sigma$: Population standard deviation.

   2. For an infinite population:

      $$ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} $$

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4. Shape of Sampling Distribution

   1. Normal Population:
      1. If the population is normally distributed, the sampling distribution of $\bar{x}$ is also normally distributed.
   2. Non-Normal Population:
      1. If the population is not normally distributed, the sampling distribution of $\bar{x}$ approaches normality as $n$ becomes large ($n > 30$) by the Central Limit Theorem.

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5. Key Properties of Sampling Distribution

   1. Reduction in Standard Error:
      1. As $n$ increases, $\sigma_{\bar{x}}$ decreases, indicating higher precision in the sample mean.
   2. Diminishing Returns:
      1. Increasing $n$ reduces $\sigma_{\bar{x}}$, but the rate of improvement diminishes.

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6. When to Use the Sampling Distribution Formula

   1. Recognizing Sampling Distribution Problems:
      1. Key Clue 1: The problem mentions sample mean or sample proportion.
      2. Key Clue 2: The problem involves sampling (e.g., "a random sample of size n is taken from a population").
      3. Key Clue 3: You're tasked with finding probabilities related to the sample mean ($\bar{x}$) or how the mean varies from sample to sample.
   2. Common Scenarios:
      1. Problems that require calculating the standard error ($\sigma_{\bar{x}}$) or understanding how the sample mean distribution behaves.
      2. Words like "sample size," "expected value of sample mean," or "distribution of sample means" appear.

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