Sampling and Sampling Distributions: Exhaustive Study Guide

Introduction to Inferential Statistics

Inferential statistics consist of statistical methods that utilize sample statistics to estimate population parameters of random variables currently under study.
The process involves moving from descriptive statistics (summarizing a sample) to inferential statistics (making predictions or conclusions about the entire population).
Real-World Example: A quality controller calculates the mean mass of a sample of $130$ cornflakes boxes. This calculation is a descriptive statistic. The controller then uses this result to determine if the filling machine is filling all cornflakes boxes to the correct specification mass, which constitutes inferential statistics.
The four pillars supporting inferential statistics are:
- Descriptive statistics: Including sample mean ( $\bar{x}$ ), proportion ( $p$ ), and standard deviation ( $s$ ).
- Probabilities: Specifically focusing on the normal distribution.
- Sampling methods: Understanding their influence on sampling error.
- The concept of the sampling distribution.

Sampling and Sampling Methods

Challenges in Populations: When studying a population, it is often impossible to access all $N$ units due to constraints in time, cost, and general accessibility.
The Solution: Researchers randomly select a representative sample of size $n$ and calculate sample statistics. These statistics serve as the basis for making inferences regarding the corresponding population parameters.
Validity Requirements: To produce valid and reliable estimates, the sampling method used to collect data must be representative of the target population.
Basic Approaches: There are two primary sampling approaches:
- Non-probability sampling.
- Probability sampling.

Non-Probability (Non-Random) Sampling

Definition: Includes all sampling methods where units are not selected at random.
Applications: Useful in exploratory research or less scientific surveys to provide initial insights into random variables.
Limitations and Bias:
- These samples are unlikely to accurately represent the target population.
- They introduce bias into statistical findings.
- It is impossible to measure sampling error (the difference between the actual population parameter value and its sample statistic).
- Statistical inferences drawn from non-probability sample data are considered invalid.
Types of Non-Probability Sampling:
- 1. Convenience Sampling: The researcher selects units that are readily available. Example: Interviewing motorists at only one petrol station regarding fuel brand preferences because they are easy to find.
- 2. Judgment Sampling: Researchers use their own judgment to select units they believe are best. Example: Selecting only labor union leaders to respond to a study on mining industry working conditions, rather than general workers.
- 3. Quota Sampling: Quotas are set for specific subgroups of a population. Once a quota (e.g., $20$ males and $20$ females) is reached, no more units from that subgroup are selected.
- 4. Snowball Sampling (Chain-Referral): Initial units refer the researcher to other units. This is used for sensitive or hard-to-locate populations. Example: Research involving HIV, drug addiction, or illegal immigrants.

Probability (Random) Sampling

Definition: Sampling methods where units are selected at random from the target population. All $N$ units in the population have a chance of being selected.
Benefits:
- Reduces selection bias.
- Likely to produce unbiased estimates.
- Sampling error can be measured, making these samples suitable for inferential statistics.
Types of Probability Sampling:
- 1. Simple Random Sampling: Every unit has an equal chance of being selected. This is appropriate for a population that is homogeneous regarding the variable under study (e.g., selecting Lotto numbers).
- 2. Systematic Random Sampling: Used when a sampling frame (list/database) exists. The first unit is selected randomly, then units are selected at uniform intervals of size $k$ . Steps include:
- Determine interval: $k = \frac{N}{n}$
- Randomly select a starting point from $1, 2, \dots, k$ within the first block.
- Select every $k$ th unit thereafter.
- 3. Stratified Random Sampling: Used when a population is heterogeneous. The population is divided into homogeneous segments called strata. Simple random samples are drawn from each stratum proportional to the stratum's size.
- 4. Cluster Random Sampling: Used when a population naturally divides into clusters that have similar profiles. A sample of clusters is randomly selected. Units within these clusters may then be randomly selected to provide a representative sample.

Comparisons of Sampling Methods

Simple vs. Stratified Shopping:
- Example: A sample of $200$ households in Gauteng needs to include all socio-economic classes (SEC).
- Simple random sampling might accidentally exclude, under-represent, or over-represent a specific SEC.
- Stratified sampling uses population data on SEC distribution to create strata, then selects households randomly from each stratum to reflect the relative distribution.
Quota vs. Stratified:
- While both use subgroups, the key difference is the selection methodology (non-probability for Quota, probability for Stratified).
- In Quota sampling, units are selected based on observation until the quota is filled.
- In Stratified sampling, sample points are selected so strata are naturally represented, and all selected units must be included.
Stratified vs. Cluster:
- Stratified: Researchers are interested in all strata and take a sample from each.
- Cluster: Researchers select a random sample of clusters/strata first, then sample units within only those chosen clusters.

Educational Exercises

Exercise 1.1: Identify Sampling Methods

1. Selecting a random sample of $25$ out of the $371$ active telephone area codes in South Africa.
2. At a party with $30$ students aged $21+$ and $15$ students under $21$ , selecting a sample of $size \, 6$ to measure attitudes toward alcohol.
3. Tree species count in a $20\text{m} \times 25\text{m}$ rectangle within a large forest in KZN.
4. Sample of $10$ houses in a neighborhood of $400$ households.
5. Calculation: A stratified random sample of $230$ consumers from four areas (A, B, C, D) with a population distribution of $5:3:1:1$ . How many people are chosen per area?
6. Calculation: A forest has $50$ sections of similar biodiversity, each divided into four areas. A biologist uses cluster sampling to select $20$ areas total, taking two from each section. What percentage of sections did he choose?

Concept of the Sampling Distribution

Purpose: Sample statistics vary from sample to sample. To assess the reliability of an inference, one must know the probability distribution of the statistic used (the sampling distribution).
Definition: The probability distribution of a particular sample statistic.
Notation Table:

Measure	Sample Notation	Population Notation
Mean	$\bar{x}$ (x-bar)	$\mu$ (mu)
Variance	$s^2$ (s-squared)	$\sigma^2$ (sigma-squared)
Standard Deviation	$s$	$\sigma$ (sigma)
Proportion	$p$	$\pi$ (pi) [Note: not $3.1416$ ]
Size	$n$	$N$

Deriving the Sampling Distribution of the Mean

Theory: If a random sample of size $n$ is selected and the mean ( $\bar{x}$ ) calculated, and this process is repeated an infinite number of times, the resulting list of averages forms a new random variable called $\bar{X}$ .
Characteristics:
- $\bar{X}$ has its own probability distribution, mean (expected value), and variance/standard deviation.
- This distribution depends on the distribution of the original variable $X$ (the parent population).
Normality Conditions:
- If the original variable $X$ is normally distributed, $\bar{X}$ is normally distributed.
- If the original variable is not normally distributed or unknown, $\bar{X}$ is normally distributed if $n \ge 30$ (Central Limit Theorem).

The Central Limit Theorem (CLT)

Definition: Regardless of the shape of the underlying population from which a sample is drawn, as the sample size ( $n$ ) increases, the sampling distribution of the mean (or proportion) approaches the normal distribution.
Parameters of the Sampling Distribution of the Mean:
- The mean of the sampling distribution equals the mean of the original variable: $E(\bar{X}) = \mu$ .
- The standard deviation of the sampling distribution is called the Standard Error.
- Distribution notation when $\sigma$ or $\sigma^2$ is known: $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$ .
- If distribution is unknown but $n \ge 30$ : $\bar{X} \sim N(\mu, \frac{\sigma^2}{n})$ .

Sampling Distribution of a Proportion

Sample proportion ( $p$ ) is used to make inferences about the population proportion ( $\pi$ ).
Logic for Derivation: Similar to the mean, a sampling distribution can be derived for proportions.
Parameters: $p \sim N(\pi, \frac{\pi(1-\pi)}{n})$ .
Normality Condition: Provided that $n\pi \ge 5$ and $n(1-\pi) \ge 5$ (per CLT).

Exercise 1.3: Derivation Practice

1. Height of females ( $X$ ) follows $N(165, 10^2)$ . Derive the sampling distribution of the mean for $n = 25$ females.
2. Average lightbulb lifetime is $80$ days with a standard deviation of $5$ days. Derive the sampling distribution of the mean for $n = 49$ bulbs.
3. Use the lightbulb information to derive the sampling distribution of the mean for $n = 20$ bulbs.
4. Research indicates $65\%$ of motorists wear seatbelts. For a sample of $100$ motorists, derive the sampling distribution of the proportion ( $p$ ) of motorists wearing seatbelts.

Inferential statistics help us make conclusions about a large group (population) by studying a smaller part of it (sample). For example, if you want to know if a type of cereal has proper weight, you could weigh a few boxes instead of every single one. The average weight of the few boxes tells you about the whole lot, which is called making an inference.

There are four key things in inferential statistics: 1) Descriptive statistics (this is about summarizing the data you collect), 2) Probabilities (understanding how likely something is), 3) Sampling methods (how you pick your smaller group), and 4) The concept of sampling distributions (how much the averages can vary when you take samples).

When collecting data, we often can't look at everybody because it's too hard or impractical, so we take samples. The way we choose these samples can be either random (everyone has a fair chance of being picked) or not. Random sampling generally gives us better and more accurate information about the population whereas non-random might lead to biases and inaccurate conclusions. There are many types of sampling methods, such as picking people who are easy to find (convenience) or making a plan to make sure everyone is represented (stratified).

Overall, inferential statistics helps us to make educated guesses about a big group based on smaller pieces of it, provided we select our samples carefully and think critically about what they mean.