Sampling Methods and Distributions

Sampling and Sampling Distributions Overview

• Chapter 3 focuses on sampling methods and sampling distributions.
• Chapter 1 introduced descriptive statistics using numerical and graphical methods.
• Chapter 2 extended statistical inference using probability and probabilistic distributions.
• Binomial and Poisson distributions (discrete) and the normal distribution (continuous) are highlighted.
• Probability distributions encompass all possible experiment outcomes and their associated probabilities.
• Probability distributions are used to assess the likelihood of future events.
• Statistical inference aims to determine population characteristics from a sample.
• The population refers to the whole group of individuals or items under consideration.
• A sample constitutes a subset of the population.
• This chapter introduces sampling and methods for selecting a sample from a population.
• Construction of the sample mean and proportions is discussed to understand how sample means cluster around the population mean.

Sampling Methods

• Inferential statistics helps find and generalize information about a population using a sample.
• A sample is a portion of the population of interest.
• Sampling offers a more feasible approach compared to studying an entire population.

Reasons to Sample

• Studying characteristics of a population often involves selecting samples.
• Sampling can be more practical for observing and measuring than studying the entire population due to:
• The time needed to contact the whole population can be prohibitive.
• Studying all items in a population can be too costly.
• Physically checking all items may be impossible, especially with large, mobile, or short-lived populations (e.g., fish, birds, mosquitoes).
• Some tests are destructive, making it impossible to test all items (e.g., wine tasting, tensile strength testing of wires).
• Adequate sample results means a 100% sample isn't essential for most problems.
• For example, the Ethiopian government uses a sample of grocery stores across the country to determine the monthly food price index as including every store wouldn't drastically change the index.

Random Selection

• Random selection constitutes a designed selection process guaranteeing every population element a known, non-zero chance of inclusion in the sample.
• When selecting a sample, researchers should ensure fair population representation.
• Ethical statistics requires unbiased sampling and objective reporting.
• Two broad classes of sampling methods exist; random/probability sampling methods and non-random/non-probability sampling.

Simple Random Sampling

• Simple random sampling involves selecting a sample where each item/person in the population has an equal chance of inclusion.
• To illustrate, a population of 845 Nitra Industries PLC employees needs a sample of 52.
• Give each employee an equal chance of selection by writing each name on a slip of paper.
• Deposit the slips in a box, mix them thoroughly, and draw slips randomly until 52 employees are selected.
• A more convenient method involves using employee identification numbers and a random number table.
• Random numbers have an equal probability for each digit (0-9); therefore, employee 011, 722, and 382 have the same selection probability.
• Random numbers eliminate bias.

Systematic Random Sampling

• Can be more appropriate than simple random sampling in some research.
• Computer Printers Unlimited requires to estimate mean dollar revenue per sale quickly.
• They decide to select 100 invoices from 2,000 recorded invoices.
• It would be time-consuming to number each invoice and use a random number table for simple random sampling.
• Systematic random sampling involves selecting a random starting point and then selecting every kth member of the population.
• Calculate k by dividing the population size by the sample size.
• For Computer Printers Unlimited, the 20th (2,000/100) invoice from the file drawers is selected, which avoids the numbering process.
• If k is not a whole number, round down.
• Simple random sampling selects the first invoice and a number between 1 and k (or 20).
• For example, if the random number is 18, then the 18th invoice, and every 20th invoice from there (38, 58 etc.), would be selected.
• Before using systematic random sampling, the physical order of the population must be carefully observed.
• Don't use systematic random sampling if the physical order relates to the population characteristic because systematic random sampling would not guarantee a random sample.

Stratified Random Sampling

• Stratified random sampling helps represent each group in a population that can be divided into distinct groups based on certain characteristics.
• These groups represents "strata", such as dividing college students by full-time/part-time status, gender or traditional status.
• Simple random sampling can be applied within each stratum to collect the sample after defining each stratum.
• Stratified random sampling means dividing a population into subgroups (strata), then randomly selecting a sample from each stratum.
• To study of the advertising expenditures for the 352 largest companies in the United States, the goal is to find if firms with high returns on equity spend more per sales dollar on advertising than firms with a low return or deficit.
• Companies are grouped by percent return on equity to ensure a fair representation of the 352 companies.
• A table (3-1) represents the strata and relative frequencies.
• Simple random sampling would give firms in the 3 and 4 strata a high chance of selection.
• Stratified random sampling guarantees at least one firm selected from strata 1&5.
• Out of 50 firms selected for study, the number of firms sampled from each stratum is proportional to the stratum’s relative frequency in the population.
• Stratified sampling can accurately reflect population characteristics compared with simple or systematic random sampling, particularly helpful when the population can be divided into two or more heterogeneous groups.

Cluster Sampling

• Cluster sampling reduces the cost to sample
• Cluster sampling divides a population into homogeneous groups using naturally existing geographic or other boundaries (clusters).
• Clusters are randomly selected; then, a sample is collected by randomly selecting from each cluster.
• To learn the views of the residents of Amhara Regional State about Ethiopian governmental environmental protection policies is cluster sampling.
• Cluster sampling subdivides the state into smaller units like weredas, also called primary units.
• By selecting four regions randomly such as regions 2, 7, 4, and 12 and focusing efforts here
• Taking a random sample of the residents in each of these regions and interviewing them (a combination of cluster sampling and simple random sampling.)

Sampling "Error" and Sampling Distribution of the Sample Mean

• Sampling methods that can fairly represent the population.
• The selection of every possible sample of a specified size from a population has a known chance or probability
• Samples estimate population characteristics such as estimating population mean
• Due to the samples being a portion of the population, the sample mean would likely NOT be the population mean
• A difference is expected between a sample statistic and the corresponding population parameter termed sampling error.

Sampling Error Example

• Tartus Industries PLC (seven employees) represents a population with hourly earnings from Table 3-2. To define its sampling error.
• Key Questions:
  • What are the parameters: the population mean and standard deviation?
  • What is the sampling distribution of the sample mean for samples of size 2?
  • What is the mean and standard deviation (Standard Error) of the sampling distribution?
  • What observations can be made about the population and the sampling distribution?
• Population mean is calculated as the sum of wages ($7+$7+$8+$8+$7+$8+$9) divided by 7 = $7.71
• Population parameters use Greek letters (µ for the mean.)
• Greek letter Sigma (σ) represents standard deviation: 0.70.
• To get sampling distribution of the sample mean, all 2 samples are selected.
• Samples found by using formula (N/n) = N! / n! (N-n)!.
• With a sample of 2 = 7!/(2!)(7-2!) = 21.
• In Table 3-4, sampling distribution replaces frequency table (Table 3-5). The table computes of the mean and standard deviation (standard error),
• Results:
  • (1) Population Parameters.
  • (2) Sampling distribution for samples,
  • (3) sampling table to computer Mean
• Population mean= mean distribution (7.71) µx=µ
• Standard deviation of the population σ > σẋ

Estimating the Standard Error

• Formula for estimating the standard error of the mean includes population size, population mean, and standard deviation using:

σ x= σ/√ n √N-n/N-1

• In cases where the sample is less than 5% of the population eliminates the right side of the previous formula
• Formula to estimate the standard error of the mean given the population standard deviation and the desired sample size:
• σ x = σ/√ n

Central Limit Theorem

• The central limit theorem and it's use in creating confidence intervals and perform tests of hypothesis.
• The central limit theorem: for large random samples, the shape of the sampling distribution of the sample mean is approximately a normal probability distribution.
• Conclusions:
  • Large samples = accurate findings
  • We can understand about sample mean no matter the shape of the distribution (true for all distributions)
• Central Limit Theorem: size samples approximately equal a normal distribution improving with sample size.
• A population and distribution will result in a normal sampling distribution of the sample mean for any sample size.
• Symmetrical (but not normal) population, the normal shape distribution will happen with samples = 10
• Thick-tailed distributions need samples of 30+ to see normality features.
• Summary: the theorem moves distribution to more observations and stronger convergence