AP Statistics: Unit 5 - Sampling Distributions

Parameter

A number that describes a characteristic of the population.

Statistic

A number that describes a characteristic of a sample.

Sampling Distribution

The distribution of values taken by a statistic in all possible samples of the same size from the same population.

Unbiased Estimator

A statistic is unbiased if the mean of its sampling distribution equals the true value of the parameter.

Bias

Failure of the sampling distribution to center on the population parameter.

Variability

Describes how spread out the values of the sample statistic are.

Shape of Sampling Distribution

Approximately Normal if the Large Counts Condition is met.

Mean of Sample Proportion

For the sample proportion, the mean is equal to the population proportion (μₓ = p).

Standard Deviation of Sample Proportion

Calculated as σₓ = sqrt[p(1-p)/n].

10% Condition

The sample size must be less than 10% of the population size.

Large Counts Condition

Both np and n(1-p) must be greater than or equal to 10 for the sampling distribution to be approximately Normal.

Center of Sampling Distribution for Differences in Proportions

The difference in population proportions, p₁ - p₂.

Spread of Sampling Distribution for Differences in Proportions

Calculated as σₓ₁₋ₓ₂ = sqrt[(p₁(1-p₁)/n₁) + (p₂(1-p₂)/n₂)].

Sampling Distribution for Sample Means

Uses the sample mean x̄ to estimate the population mean μ.

Standard Deviation of Sample Mean

Calculated as σₓ = σ/sqrt(n).

Central Limit Theorem (CLT)

States that the sampling distribution of the mean will be approximately Normal if the sample size is large enough (n ≥ 30).

Random Sample

Each individual in the population has an equal chance of being included in the sample.

Normality Condition

The sampling distribution of the sample mean is Normal if the population is Normal or by the CLT if the sample size is large.

Independent Samples

Samples are independent if the sample selections do not influence each other.

Notation for Sample Mean

x̄ represents the sample mean.

Notation for Population Mean

μ represents the population mean.

Standard Error

The standard deviation of the sampling distribution of a statistic, often denoted as SE.

Example: Estimate of High Math Anxiety

If p = 0.80 and n = 110, what are the mean and standard deviation of the sampling distribution of the sample proportion?

Sample Size Effect on Variability

Larger samples yield smaller variability in estimates.

Probability Calculation Command on TI-84

Use normalcdf() for computing probabilities under the curve.

Finding Percentiles Command on TI-84

Use invNorm() to find cut-off values based on probabilities.

Mistake: Law of Large Numbers vs. CLT

LLN is about sample averages approaching the population mean; CLT is about the shape becoming Normal.

Population Proportion Notation

Use p for the population proportion and use p̂ for the sample proportion.

Standard Deviation vs. Standard Error Confusion

Standard deviation is for population; standard error is for sampling distributions.

Independent Random Samples Condition

Condition that both samples must be selected independently.

Mean of Sampling Distribution for Sample Means

The mean of the sampling distribution equals the population mean (μ).

Standard Deviation of Sampling Distribution for Sample Means

Calculated as σ/sqrt(n) for sample means.

Random Sampling Condition

The sample must be drawn in such a way that every member of the population has an equal chance of selection.

Hypothesis Testing Basics

Involves comparing sample statistics against known parameters to infer properties about populations.

Two Sample t-Test

A test used to compare the means of two independent samples.

Sample Size Implication on Estimates

Increasing sample size generally leads to more reliable estimates of population parameters.

Normal Approximation Requirement in Sampling Proportions

Requires that np and n(1-p) are both greater than or equal to 10.

Variances of Independent Samples

The variances add together, but standard deviations do not when calculating spread of differences.

P-value in Hypothesis Testing

Indicates the probability of obtaining test results at least as extreme as the observed results under the null hypothesis.

Bias Correction in Sampling

Adjusting estimates to reduce system bias in sample data.

Random Sampling Importance

Ensures that estimates are generalizable and unbiased.

Sampling With Replacement

Each selected individual is returned to the population before the next selection.

Sampling Without Replacement

Selected individuals are not returned for future selection, affecting independence.

Confidence Interval Basics

A range of values used to estimate a population parameter.

Z-score Calculation

Used to determine how many standard deviations an element is from the mean.

Sampling Framework Importance

Defines how we apply statistical methods effectively to sample data.