Sampling Overview

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1. Sampling
   1. Sampling Theory
   2. Principles of Sampling
   3. Sampling Methods
   4. Simple Random Sampling </aside>

Sampling Theory

1. Sampling Theory

   1. Sampling is a process where a subset of a population, called a sample, is selected to collect data for analysis. Sampling is fundamental in statistics as it allows us to infer characteristics of the entire population without having to analyze every individual element. Below is a detailed breakdown of the concepts involved:

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2. Key Concepts

   1. Population and Sample
      1. Population:
         1. Definition: The complete set of elements (individuals, objects, or data) of interest in a particular study.
         2. Example: All users of a mobile application, all students in a school, or all households in a city.
         3. Characteristics measured from a population are called parameters (e.g., population mean $\mu$, population standard deviation $\sigma$).
      2. Sample:
         1. Definition: A subset of the population selected for analysis.
         2. Example: A survey conducted on 500 out of 10,000 app users.
         3. Characteristics measured from a sample are called statistics (e.g., sample mean $\bar{x}$, sample standard deviation $s$).
   2. Population Parameters vs. Sample Statistics

      1. Parameters and statistics correspond to the same type of measurement but apply to the population and sample, respectively:

         Measure of Characteristic	Population Parameter ($\mu, \sigma, p$)	Sample Statistic ($\bar{x}, s, \hat{p}$)
         Mean	$\mu$	$\bar{x}$
         Standard Deviation	$\sigma$	$s$
         Proportion of Success	$p$	$\hat{p}$

      2. Example:

         1. Population Parameter: The average score of all students in a school.
         2. Sample Statistic: The average score of a sample of 50 students selected from the school.
   3. Census vs. Sample Survey
      1. Census:
         1. A survey that collects data for the entire population.
         2. Example: A national population and housing census.
      2. Sample Survey:
         1. A survey that collects data from a sample to infer characteristics of the population.
         2. Example: Conducting an opinion poll of 1,000 voters to estimate national voting preferences.
   4. Estimation in Sampling
      1. Point Estimation:
         1. Estimating a population parameter using a sample statistic.
         2. Example: Using the mean of a sample ($\bar{x}$) to estimate the population mean ($\mu$).
      2. Point Estimator:
         1. The formula or rule that provides the point estimate.
         2. Example: For population mean $\mu$, the sample mean $\bar{x}$ is the point estimator.
      3. Point Estimate:
         1. The specific value calculated from a sample.
         2. Example: If the sample mean is $\bar{x} = 36,500 \, \text{km}$, then this is the point estimate for the population mean.
   5. Types of Populations
      1. Finite Population:
         1. The population consists of a limited number of elements.
         2. Example: A company surveys 5,000 out of 100,000 app users about a new feature.
         3. Inference: Based on the sample, the company estimates that 77.9% of all users will like the feature.
      2. Infinite Population:
         1. The population is too large to count or theoretically infinite.
         2. Example: A tire manufacturer tests 120 tires to estimate the average lifespan of all tires it produces.

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3. Example Problems

   1. Example 1: Finite Population
      1. Problem:
         1. An app developer introduces changes to an app for 5,000 out of its 100,000 users. From the 5,000 surveyed users, 3,895 liked the changes. Based on this data, estimate the proportion of the entire population that will like the changes.
      2. Solution:

         1. Sample Size ($n$):

            $$ n = 5,000 $$

         2. Sample Proportion ($\hat{p}$):

            $$ \hat{p} = \frac{\text{Number of users who liked the changes}}{\text{Total number of surveyed users}} = \frac{3,895}{5,000} = 0.779 $$

         3. Inference for Population Proportion ($p$): Assuming the sample is representative:

         4. Answer: Based on the sample, approximately $77.9\%$ of all users will like the changes.

   2. Example 2: Infinite Population

Principles of Sampling