Statistics for Business - Sampling, Estimation

STATISTICS FOR BUSINESS

SAMPLING AND ESTIMATION

Introduction to Statistics

• Definition of Statistics: Statistics involves the study of phenomena occurring en masse.

• Statistical Population:

  • Definition: All the units (items) being the object of a study; collection of all elements of interest.

  • Types of Population:

  • Discrete Population: Countable units.

  • Continuous Population: Measurable data.

  • Finite Population: Limited number of units.

  • Infinite Population: Unlimited units.

Data Collection Methods

• Full Scope Data Collection:

  • Definition: Observing all elements of the population.

  • Features:

  • Time-consuming.

  • Costly.

  • Applicable for both infinite or finite populations.

• Sample-Based Data Collection:

  • Definition: Observing only a subset of elements from the population.

  • Related to inferential statistics, where conclusions are drawn about the population based on sample data.

Sampling

• Sample Definition: A set of elements chosen (drawn) from the population.

• Sampling Frame: A reference list of the population, which is always finite.

  • Population Characteristics:

  • Probability samples rely on randomness, which ensures reliability and allows for error calculation.

  • Sample elements are represented as random variables, and their values differ sample by sample, denoted as (X₁, X₂, …, Xn).

• Sample Size (n):

  • Example size: n = 5.

Sample Planning

• Considerations for Sampling:

  • Balancing accuracy versus costs/time in sample size.

  • Types of the population: infinite or finite.

  • Types of sampling techniques (with/replacement).

• Practical Consideration: Large finite populations may be considered as infinite in practice, simplifying sampling techniques.

Sampling Errors

• Definition of Errors:

  • Sampling Errors: Errors that depend on the chosen sample, which can be measured and calculated.

  • Non-Sampling Errors: Difficult to measure, includes:

  • Frame errors (undercoverage or overcoverage).

  • Non-response (full or partial).

  • Measurement errors (imprecise information).

  • Processing errors (errors in data capturing or units measurement).

  • Mode effect caused by the selection method.

• Random Selection: Ensures a random (probability) sample.

Sampling Methods

• Independent & Identically Distributed Sampling (IIDS):

  • Also known as Random Sampling.

  • Definition: From homogeneous, infinite populations with or without replacement.

  • Key Note: Elements in the sample are independent from each other; however, practical independence is challenging to achieve.

  • Application: Common in quality control in production.

• Simple Random Sampling (SRS):

  • Definition: Sampling from a homogeneous, finite population without replacement.

  • Requirements: A complete list of population elements and equal probability of selection for each element.

• Stratified Sampling (SS):

  • Utilizes heterogenic populations divided into strata, applying SRS within each stratum.

  • Objectives: Aim to produce more accurate estimations, reducing errors.

  • Types of Stratified Sampling:

  • Proportionate Allocation: Sample sizes according to strata proportion.

  • Disproportionate Allocation: Variations in allocations based on stratum characteristics.

  • Optimal Allocation: More sample elements from strata with higher variability.

  • Cost-Optimal Allocation: Minimizing error at certain cost levels.

• Cluster Sampling (CS):

  • Used when no adequate sample frame exists.

  • Method: Population is grouped, SRS is drawn from these groups (clusters), and then complete observation within the selected clusters.

  • Variations: Multistage sampling, applying SRS multiple times.

• Systematic Sampling (SYS):

  • Population is arranged according to a certain aspect, and a starting point and interval (k) are determined.

  • Process: Selecting every kth element after the first selected element from the sorted population.

  • Cautions: Ensures that no stochastic relationship exists between the sorting variable and the observation.

• Non-Probability Sampling: Anticipates a systematic sampling when relationships exist between variables. Types include:

  • Quota Sampling: Fixed composition with random selection.

  • Snowball Sampling: Selected elements refer others in their network.

  • Concentrated Sampling: Influential elements have a higher selection probability.

  • Judgment Sampling: Often used in public opinion surveys.

Inferential Statistics

• Estimation: Estimating values of unknown population parameters using sample data.

• Hypothesis Testing: Testing assumptions about the population (parameters or other features) with sample data.

Basic Concepts of Estimation

• Parameter (θ): Numerical characteristics of a population such as expected value E(X) = μ, standard deviation D(X) = σ, and proportion P.

  • Parameters are constant values; the goal is to estimate them accurately.

• Estimator (θ̂): A rule for calculating an estimate based on observed data.

• Sample Statistic: Characteristics calculated from the sample to estimate parameters.

• Point Estimator: The statistic that yields a point estimate for a parameter (e.g., mean x̄, standard deviation s, proportion p).

• Sampling Distribution: A probability distribution of all sample statistics.

• Standard Error: The standard deviation of a point estimator often denoted as σx, S_x, S_p.

Properties of a Point Estimator

• Unbiased: The expected value equals the parameter estimated.

• Asymptotic Unbiasedness: Difference reduces as sample size increases.

• Consistency: As sample size increases, the estimator approaches the true parameter value.

• Effectiveness: Smallest standard error among competing estimators.

• Sufficiency: Contains all information about characteristics from the population.

Theorems in Statistics

• Law of Large Numbers: As the number of trials increases, the average result approaches the expected value.

• Central Limit Theorem (CLT): Sum of a large number of independent random variables approximates a normal distribution (bell curve).

• Chebyshev's Theorem: A maximum fraction of values will not be more than a specified distance (in terms of standard deviations) from the mean.

Interval Estimation

• Interval Estimation Definition: An estimate providing a range (interval) believed to contain the parameter with a certain confidence level (π).

• Formula: Point estimate ± Margin of error [L, U], where P(L < Parameter < U) = π.

• Factors Affecting Width of Confidence Interval:

  • Sample Size (n).

  • Standard Deviation of the estimator.

  • Confidence Level (π).

  • Distribution of the population.

  • Maximum error of estimation (A).

Estimation Processes

• Estimating Expected Value:

  • For populations having unknown distributions:

  • If known (σ): ar{x} ext{ ± } Z_{1-α/2} imes rac{σ}{ ext{√n}}

  • If unknown (using S): ar{x} ext{ ± } t_{1-α/2} imes rac{S}{ ext{√n}}

• Standard Error: Denoted as S_x ext{ or } rac{σ}{ ext{√n}}.

Estimation of Probability (Proportion)

• Probability Estimator:

  • Point estimate: ext{p} = rac{k}{n}

  • Standard Error (Sp): Sp= ext{√} rac{p imes q}{n} where q = 1-p.

Estimation of Standard Deviation

• Point Estimate: Standard deviation (s).

• Population with Normal Distribution: Uses the chi-square distribution to find confidence intervals.

• Formula for confidence intervals for s:

  • rac{(n-1)s^2}{χ{1-α/2}^{2}} ext{ and } rac{(n-1)s^2}{χ{α/2}^{2}}

Estimation from Individual and Grouped Data

• Individual Data Calculation: Utilizes functions like AVERAGE and STDEV.S in Excel or manual calculations.

• Grouped Data Calculation: Includes weighted averages and standard deviations for grouped outcomes.

• Example: Given data of usage from a sample, calculating averages and standard deviations from frequency distributions ensures accurate statistical representation.

Estimation of Sum of Value

• Application: For example, to estimate the net filling weights of a certain product using a sample.

  • Total and average calculations formulae regarding the sample and population figures are explained.

• Steps: Calculation involves:

  • ext{Average Weight} = rac{ ext{Σ (weight imes frequency)}}{n}

  • Estimating total population weight based on sample average.

Determining Sample Size

• Margin of Error Definition: Refers to the appropriate range for confidence estimation.

• Formulae for Sample Size:

  • Sampling distributions involve critical values from the standard normal distribution and standard errors.

  • Careful calculation of sample size based on desired estimation accuracy and existing sample size is critical.

Example Calculation for Sample Size Increase

• If an original sample of size 120 aims to reduce estimation error by 4%, the adjusted sample size calculation considers the margin of error reduction as well as confidence intervals.