Notes 7-1

7.1 Sampling Distributions

Quick Review of Parameters vs Statistics

• Parameter

  • Definition: A number that describes some characteristic of a population.

  • Examples:

    • Mean (M)

    • Proportion (p)

    • Population Size (N)

• Statistic

  • Definition: A number that describes some characteristic of a sample.

  • Examples:

    • Sample Mean (x)

    • Sample Proportion (k)

    • Sample Size (n)

• Understanding the differences between parameters and statistics is crucial for sampling and analyzing data.

Example 1: The Gallup Poll

• Population: All US adults.

• Parameter: Proportion of all US adults who believe in ghosts (p).

• Sample: 515 respondents from the Gallup Poll.

• Statistic: Proportion of sample who believe in ghosts:

  • Calculation: 160 said "Yes" → Statistic = 160/515 = 0.31.

Example 2: Arizona Iced Tea Quality Control

• Claim: Bottles supposed to contain an average of 20 oz of tea.

• Sample: 50 bottles taken at random.

• Statistic: Sample average = 19.6 oz.

• Parameter: Population mean (M) = 20 oz.

Sampling Variability

• Definition: The variability in statistics obtained from different random samples of the same size from the same population.

Example 3: Sampling Height

• Activity: Sampling heights from the class using samples of size 4.

• Task: Each student calculates the average height from random samples.

• Visualize results with a dot plot.

Page 2: Sampling Distribution

Definition of Sampling Distribution

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

• Importance: Too complex to take all samples; simulations are often used to approximate sampling distributions.

Example 4: M&M Color Proportions

• Claim: Mars estimates the colors in M&Ms as follows: 24% blue, 20% orange, etc.

• Sampling: Examine the proportion of orange M&Ms by taking repeated samples of 50.

• Note: The resulting graph should show proportions consistent with the claim.

• Understanding the Statistic: The statistic represents the sampled percentage.

Definition of Unbiased Estimator

• Unbiased Estimator: A statistic used to estimate a parameter where the mean of its sampling distribution is equal to the parameter value.

Discussion on Spread of Sampling Distribution

• Lower variability is better.

• Higher sample sizes decrease variability.

• Variability is often not significantly affected by the population size, as long as the sample size is < 10% of the population.

Page 3: Example 5 - Comparing Sample Statistics

Eruption Intervals Analysis

• Data: Population median of eruption intervals is 75 minutes.

• Sampling: Computer simulation shows sample means from 10-size samples leading to a mean of 73.3 minutes.

Questions and Justification

1. Is the sample median unbiased? No, the sampling distribution of the sample median does not equal the population median.

2. Effect of sample size on spread: Larger samples yield smaller variability in the sampling distribution.

3. Shape of the sampling distribution: Unimodal and approximately normal with larger samples.

Visualizing Estimation Accuracy

• Illustrate bias and variability using target analogy.

  • Bias: Systematic deviation from the bull's-eye (true value).

  • High Variability: Shots dispersed widely from the bull's-eye.

Page 4: Example 6 - Day Care Center Board

Characteristics of the Population

• Members:

  • Jay: 5 children

  • Carol: 2 children

  • Allison: 1 child

  • Teresa: 0 children

  • Anselmo: 2 children

Calculating Averages

• Population Mean: M = 2 children (average).

• Create a sampling distribution for sample size 2:

  • Calculate averages for all possible samples.

Relationships between Parameter and Sampling Distribution

• Comparison: Parameter M = 2 relates closely to the average of the sampling distribution.

• Effect of Sample Size on Sampling Distribution:

  • Mean remains similar.

  • Spread decreases.

  • Shape approaches normal distribution with larger samples.

Summary of Sampling Distributions

• A sampling distribution represents values over different samples drawn from a fixed population, contrasting with the population distribution of a variable.