Lecture Notes Flashcards

Flashcards for reviewing key vocabulary and concepts related to the sample mean from lecture notes.

Sample Mean

A statistic calculated from a sample of data, used to estimate the population mean; different samples lead to different sample statistics.

Population

All units, people, or objects of interest in a research study.

Population Mean (μ)

The average of all values in the population, denoted by the Greek letter mu.

Population Standard Deviation (σ)

The standard deviation for all values in the population, denoted by the Greek letter sigma.

Sample Average (x̄)

The average of the values in a sample, used to estimate the population mean.

Statistics Convention

Use capitalized letters for random variables (e.g., X) and lowercase letters for sample realizations or outcomes (e.g., x).

Expected Value of x

Another name for the population mean (μ); the long-run average expected if values of x are drawn at random.

Finite Population (Formula for μ)

μ = (x1* + x2* + … + xN) / N, where N is the population size and xi are the population values.

Sample Mean (x̄)

The average of n sample realizations; x̄ = (x1 + x2 + … + xn) / n

Population Variance (σ² or Var(x))

The probability-weighted average of squared deviations from the mean; also the expected value of squared deviations from the mean. Described as σ squared x.

Population Standard Deviation (σ)

The square root of the population variance.

Sample Variance (s²)

The average squared deviations from the sample mean, calculated as s² = Σ(xi - x̄)² / (n - 1).

Sample Standard Deviation (s)

The square root of the sample variance.

Population Assumptions

1. xi has a common mean μ. 2. xi has a common variance σ². 3. Different observations are statistically independent (xi is independent of xj for i ≠ j).

Shorthand Notation for Distribution

xi ~ (μ, σ²), meaning xi is distributed with a common mean μ and common variance σ².

Population Mean of Sample Means

The expected value of x̄, which is equal to the population mean (μ).

σ²x̄ (Population Variance of Sample Means)

Equal to E[x̄ - μx̄]², and it turns out that it's equal to σ²/n

Standard Deviation of Sample Means

Equal to σ / √n.

Standard Error of the Sample Mean

The estimated standard deviation of the sample mean, calculated as s / √n, where s is the sample standard deviation.

Central Limit Theorem (CLT)

If the sample size is large, the sampling distribution of x̄ will be approximately normal.

Z-score in CLT

z = (x̄ - μ) / (s/√n), is distributed normally with a mean of zero and a standard deviation of one if n is large (n > 30).

Degrees of Freedom

With formula s squared equals one over n minus 1 sum of xi minus X bar squared, the divisor n-1 is called the degrees of freedom because only n-1 terms in the sum are free to vary since they are limited by this sum relationship.