BUS 310 Chapter 6 Lessons 1 & 2

Population (Universe)

All items of interest

Sample

A portion of the population

Parameter

A summary measure about a Population (uses μ, σ, N, ∏)

Statistic

A summary measure about a Sample (uses X̄, s, n, p)

μ (mu)

Population mean

X̄ (x-bar)

Sample mean

σ (sigma)

Population standard deviation

s

Sample standard deviation

σ² (sigma squared)

Population variance

s²

Sample variance

N

Population size

n

Sample size

∏ (Pi)

Population proportion

p

Sample proportion

Non-Probability Sampling

Sampling where items are chosen without regard to their probability of occurrence (includes judgment and convenience sampling)

Judgment Sampling

A non-probability method based on experts' opinions

Convenience Sampling

A non-probability method that is easy and inexpensive

Probability Sampling

Sampling based on chance; includes simple random, systematic, stratified, and cluster sampling

Simple Random Sample

A sample in which each population element has an equal chance of being selected, and selecting one subject does not affect selecting others

Systematic (Periodic) Sample

A sample formed by dividing the population into k groups (k = N/n), randomly selecting one individual from the first group, then selecting every kth individual thereafter

Systematic Sampling: k formula

k = N / n, where N = population size and n = sample size

Systematic Sampling Risk

Bias may occur due to periodicity

Stratified Sample

A sample formed by dividing the population into mutually exclusive, exhaustive subgroups (strata) sharing a common characteristic, then taking a simple random sample from each stratum proportional to its size

Cluster Sample

A sample formed by dividing the population into representative clusters, randomly selecting clusters, then sampling all or some items within those selected clusters

Cluster Sampling Example

Election exit polls, where certain election districts are selected and sampled

Stratified Sampling Example

Sampling a population of voters, stratifying across racial or socio-economic lines

Advantage of Simple Random/Systematic Sampling

Simple to use

Disadvantage of Simple Random/Systematic Sampling

May not be a good representation of the population's underlying characteristics

Advantage of Stratified Sampling

Ensures representation of individuals across the entire population

Advantage of Cluster Sampling

More cost effective

Disadvantage of Cluster Sampling

Less efficient — needs a larger sample to acquire the same level of precision

Estimation

Assessing the value of an unknown population parameter using sample data

Estimators

Measures used to estimate population parameters (e.g., sample mean, sample variance, sample proportion)

Point Estimate

A single number derived from sample data used to estimate the value of a population parameter

Sampling Error

The difference between the point estimate and the true value of the population parameter being estimated

Effect of Sample Size on Sampling Error

As the sample size increases, the sampling error decreases

Sampling Distribution of the Mean

The distribution of the means of all possible samples of a fixed size n from some population

Standard Error of the Mean

The standard deviation of the sampling distribution of the mean; measures how much estimates vary from sample to sample

Effect of n on Standard Error

As n increases, the standard error decreases (larger sample sizes have less sampling error)

Central Limit Theorem (CLT) — Part 1

If the sample size is large enough, the sampling distribution of the mean is approximately normally distributed regardless of the population's distribution, and has a mean equal to the population mean

Central Limit Theorem (CLT) — Part 2

If the population is normally distributed, the sampling distribution of the mean is also normally distributed for any sample size

Formula: Mean of sampling distribution

μ_X̄ = μ

Formula: Standard error of the mean

σ_X̄ = σ / √n

Rule of thumb for "large enough" sample

n > 30

Confidence Interval Estimation

Uses sample data to obtain a point estimate and a confidence interval around that point estimate

Hypothesis Testing

Determines whether the observed data provide support for a particular hypothesis

Confidence Interval

An interval around the point estimate, calculated from sample data, that is very likely to contain the true value of the population parameter

Margin of Error

The amount added to and subtracted from the point estimate to form the confidence interval; equals (Critical Value)(Standard Error)

General Confidence Interval Formula

Point Estimate ± (Critical Value)(Standard Error)

Confidence Level

The probability (1 − α) that the confidence interval correctly estimates the true population parameter; commonly 90%, 95%, or 99%

Interpretation of a 95% Confidence Interval

If we chose 100 different samples leading to 100 different interval estimates, we would expect 95 of them to contain the true population mean

Confidence Interval for μ — σ Known (formula)

X̄ ± Zα/2 (σ / √n)

Assumptions for CI with σ known

σ is known; population is normally distributed (or if not, use a large sample, n > 30)

Zα/2

The value of the standard normal random variable for an upper tail area of α/2

Zα/2 for 95% confidence

1.96

Zα/2 for 90% confidence

1.645

Excel formula for Zα/2

=NORM.S.INV(1 − α/2)

Excel formula for margin of error (σ known)

=CONFIDENCE.NORM(alpha, standard_deviation, size)

Confidence Interval for μ — σ Unknown (formula)

X̄ ± tα/2 (s / √n), with df = n − 1

Assumptions for CI with σ unknown

σ is unknown; population is normally distributed (or if not, use a large sample, n > 30); use Student's t distribution

t-Distribution

A family of probability distributions with a shape similar to the standard normal distribution, distinguished by degrees of freedom (df)

Effect of df on t-distribution

As degrees of freedom increases, the t-distribution converges to the standard normal distribution

Degrees of Freedom (df) for a single-sample CI

df = n − 1

Excel formula for t-value

=T.INV(1 − α/2, n − 1)

Excel formula for margin of error (σ unknown)

=CONFIDENCE.T(alpha, standard_deviation, size)

Confidence Interval for a Population Proportion (formula)

p ± Zα/2 √[p(1−p) / n]

When to use Z vs. t for a mean's CI

Use Z when σ (population standard deviation) is known; use t when σ is unknown (use s instead)

National Motors Example (σ known)

n=50, X̄=30, σ=4 → 95% CI: 28.88 ≤ μ ≤ 31.12

National Motors Example (σ unknown)

n=50, X̄=30, s=5 → 95% CI: 28.579 ≤ μ ≤ 31.421 (df=49, t=2.0096)

Circuits Example (σ known)

n=11, X̄=2.20 ohms, σ=0.35 → 95% CI: 1.9932 ≤ μ ≤ 2.4068 ohms