7.1 - Parameters, Statistics, & Sampling Distributions

Distinction between Parameters and Statistics

• Parameter: A number that describes a characteristic of a population.

• Statistic: A number that describes a characteristic of a sample.

• Point Estimator: A sample statistic is sometimes called a point estimator of the corresponding population parameter, indicating it represents a single estimate.

Identifying Populations and Statistics

Example 1: Gallup Poll on Beliefs in Ghosts

• Population: All U.S. adults.

• Parameter: p = true proportion of all U.S. adults who believe in ghosts.

• Sample: 515 people interviewed in the Gallup Poll.

• Statistic: 𝑝̂ = proportion of sample who believe in ghosts = 160/515 = 0.31.

Example 2: Starnes Cabin Temperature

• Population: All times during a given day.

• Parameter: Minimum temperature in the cabin at all times that day.

• Sample: 20 randomly selected time readings.

• Statistic: Sample minimum temperature = 38°F.

Sampling Distributions

• Sampling Variability: Variability observed when different random samples of the same size from the same population yield different statistics.

• Sampling Distribution: Distribution of values taken by a statistic across all possible samples of the same size from the same population.

Calculating Sampling Distributions

Example: Heights of John and Carol's Sons

• Heights in inches: 71, 75, 72, 68.

• Problem:

1. List all 6 possible samples of size 2.

2. Calculate the mean of each sample; display sampling distribution of the sample mean using a dotplot.

3. Calculate the range of each sample; display sampling distribution of the sample range using a dotplot.

Evidence in Sampling Problems

Example: Mrs. Lin's Chips

• Data: Total chips claimed = 200, red chips = 100.

• Selected SRS of 20 chips; found 7 red chips (sample proportion 𝑝̂ = 7/20 = 0.35).

• Evidence Explained:

  • Two possibilities:

    1. Sampling variability occurred (truthful count of chips).

    2. Mrs. Lin might be lying (less than half the chips are red).

Simulated Sampling Distributions

• Using Technology: Simulated 500 SRSs of size n = 20 from a population of 200 chips, half red and half blue.

• Results: Analytical understanding of p̂ from simulations aids in assessing claims about parameters.

• Important Values: Presence of very low sample proportions could suggest convincing evidence against a population claim.

Statistical Terminology and Properties

• Bias: A statistic is considered an unbiased estimator if the mean of its sampling distribution equals the true parameter value.

• Variability: Lower variability in a statistic's estimates is preferred, suggesting more reliable results.

• Sample Size Effects: Increasing sample sizes generally lowers variability in estimates; however, it does not always guarantee accuracy.

• AP Exam Tip: Always specify which distribution you are discussing to avoid losing credit for ambiguous language.

Understanding Statistical Claims and Evidence

M&M'S Example

• Claim: Mix of colors is true; hypothesized proportions for various colors.

• Steps to Analyze:

  1. Identify population, parameter, sample, and statistics.

  2. Graph population distribution and sample distributions.

  3. Determine sampled proportions and evaluate against population proportions.

Summary of Understanding Sampling Distributions

• For credible estimation, utilize unbiased estimators.

• Check answers and clarify understanding of the population parameter versus sample statistics.

• When denoting sampling distributions in exercises, ensure clarity to maintain proper statistical terminology.