Chapter 7-KM
Chapter 7: Introduction to Sampling Distributions
Page 1: Introduction
Overview of Sampling Distributions
Page 2: Importance of Sampling
Decisions based on samples raise questions about results.
Many possible samples can be selected from a population.
Each sample varies, affecting accuracy and reliability.
Page 3: Sample Variability
Sample means and percentages can differ across samples.
The sampling distribution represents the distribution of all possible sample outcomes.
Understanding this distribution aids in interpreting specific sample results.
Page 4: Sampling Error
Sampling Error: Difference between a sample statistic and the population parameter.
Even with careful random sampling, a sample may not perfectly represent the population.
Page 5: Sampling Error of the Sample Mean
Focuses on the discrepancy in the sample mean compared to the population mean.
Page 6: Population Mean
Notation: µ = Population Mean
Values represented by x = Values in the population
N = Population size
Page 7: Sample Mean
Notation: x = Sample Mean
Sample values selected from the population represented as x,
n = Sample size
Page 8: Parameters and Random Samples
Parameter: Metric calculated from the entire population, remains constant if the population does not change.
Simple Random Sample: Every sample of a specified size has an equal chance of selection.
Page 9: Business Application of Sampling Error (1/5)
Case Study: Hummel Development Corporation has 12 office complexes representing a population.
Sample will influence the assessment of projects' square footage.
Page 10: Business Application (2/5)
Average area of Hummel's complexes is 158,972 square feet, representing a population parameter.
Page 11: Business Application (3/5)
Hummel is selected for new development, the client will select a simple random sample of n=5 projects for evaluation.
Page 12: Business Application (4/5)
Key selection factor: Past performance in larger projects; interested in the mean size of projects completed.
Page 13: Business Application (5/5)
Example: Sample mean is 3,900 square feet less than the population mean.
Highlights that a random sample rarely perfectly represents its population.
Page 14: Business Application (6/5)
Selecting another sample indicates variable sampling errors based on sample size and selection.
Emphasizes that varying samples can yield different sample mean results.
Page 15: Sampling Error and Sample Size (1/6)
There are numerous combinations for selecting samples; specifically, 792 samples of size 5 from 12 projects.
Page 16: Business Application (2/6)
Decision Maker typically selects one sample to estimate population efficiency.
Acceptable small errors, but large errors can mislead conclusions.
Page 17: Business Application (3/6)
Evaluates extremes in sampling error:
Smallest: -50,740 sq. ft.
Largest: +51,928 sq. ft.
Range indicates possible errors for n=5.
Page 18: Business Application (4/6)
Analyzing impacts of scaling down samples (n=3) to extreme samples.
Page 19: Business Application (5/6)
Reducing sample size increases potential sampling error ranges significantly.
Page 20: Business Application (6/6)
Larger sample sizes typically decrease error, though not a guarantee.
Average sampling error from larger samples is lower than from smaller samples.
Page 21: Summary of Sampling Error
Random samples do not perfectly represent entire populations leading to sampling error.
Sampling Distribution: Visualized as a histogram representing all possible sample means for a size.
Page 22: Sampling Distribution
Defined as distribution of all possible statistic values from random samples of set size.
Page 23: Simulating Sampling Distribution (1/4)
Case Study: Aspen Capital Management, managing client retirement funds (population size: 200).
Page 24: Simulating Sampling Distribution (2/4)
Mean of mutual fund portfolios among clients = 2.505; standard deviation = 1.503.
Page 25: Simulating Sampling Distribution (3/4)
The controller selects a sample size of 10; repeats the sample mean computation 500 times.
Page 26: Simulating Sampling Distribution (4/4)
Sample means of 500 illustrate a normal distribution despite initial skewness in the population.
Page 27: Comparing Histograms
Average of sample means approaches the population mean as samples increase.
Page 28: Sample Means Distribution
Sample mean from 500 samples equates to approx 2.41, closely mirroring the population mean of 2.505.
Page 29: Sampling Distributions - Theorem 1
Average of all sample means equals population mean, indicating unbiased estimation.
Page 30: Sample Standard Deviation
Sample standard deviation of 500 samples is .421—less than population standard deviation of 1.503.
Page 31: Sampling Distributions - Theorem 2
Standard deviation of sample means equals population standard deviation over square root of sample size (standard error).
Page 32: The Effect of Sample Size
As sample sizes increase, standard errors decrease, resulting in a more normal distribution shape.
Page 33: Sampling from Normal Distributions
Normal population leads to normally distributed sampling means with preserved population mean.
Page 34: Effect of Sample Size
Larger sample sizes yield smaller standard errors, reducing overall sampling errors.
Visualization of differences between larger and smaller sample sizes.
Page 35: Consistent Estimator
An unbiased statistic suggests it closes in on actual population parameter as sample size grows.
Sample mean is a consistent estimator for the population mean.
Page 36: z-Value in Sampling Distribution
Measures how far a sample mean deviates from population mean using standard deviation.
Page 37: z-Value and Finite Population Correction
Adjustments required when sample size exceeds 5% of population, utilizing finite population correction factor.
Page 38: Example: Scribner Products (1/5)
Manufacturing mosaic tiles; defines diagonal dimensions within specified normal distribution parameters.
Page 39: Example: Scribner Products (2/5)
Analysts sample dimensions to ensure compliance with production specifications.
Page 40: Example: Scribner Products (3/5)
First step entails determining mean tile size from sample, defining sampling distribution.
Page 41: Example: Scribner Products (4/5)
Converts sample mean to z-value to derive probabilities of larger sample dimensions.
Page 42: Example: Scribner Products (5/5)
Discrepancy indicates potential issues in production methods needing review.
Page 43: Sample Size and Distribution
Population's shape affects resulting sampling distribution, leading to normality with sufficiently large sample sizes.
Page 44: Central Limit Theorem - Theorem 4
Regardless of distribution shape, sampling means approach normal distribution as sample sizes grow.
Page 45-46: Central Limit Theorem Applications
Applies to uniform, triangular, and skewed distributions.
Page 47: Example: Westside Drive-In (1/4)
Analysis of sales indicates skewed distributions with mean orders influenced by slight variances.
Page 48: Example: Westside Drive-In (2/4)
Sample size justifies Central Limit Theorem application for determining normality in distribution.
Page 49: Example: Illustrating Probability of Interest
Develops probability scenarios around meal costs using standard values.
Page 50: Example Conclusion
Conclusions drawn emphasize the unlikelihood of sampling errors affecting established means.
Page 51: Introduction to Sampling Distribution of a Proportion
Objective to determine population proportions within various contexts across industries.
Page 52: Definitions
Sample proportion as fraction in a sample with specific attributes, and population proportion defined by corresponding numbers.
Page 53: Sampling Errors
Comparison of population proportion and sample proportion highlighting expected discrepancies.
Page 54: Proportion Case Study (1/4)
Patterson Health Clinic incident surveying patient satisfaction and data collection.
Page 55: Proportion Case Study (2/4)
Satisfaction parameter defined with actual patient survey numbers as the basis.
Page 56: Proportion Case Study (3/4)
Variability indicated with numerous potential sample combinations leading to different outcomes.
Page 57: Proportion Case Study (4/4)
Exploration of extreme outcomes illustrating the variability of sampling error.
Page 58: Distribution of Sample Proportion
Best estimate derived from sample proportions presuming certain conditions lead to binomial distributions.
Page 59: Continuation of Proportion Distribution
Page 60: Theorem on Sampling Distribution of Proportion
Sampling distributions approximate normality under specific conditions regarding the population proportion.
Page 61: z-Value Calculation for Proportions
Page 62: Coupons Case Study (1/4)
Online coupon analysis illustrates consumer trends in non-redemption rates across populations.
Page 63: Coupons Case Study (2/4)
Investigation into a specific retailer’s non-use rates and implications.
Page 64: Coupons Case Study (3/4)
Establishing norms considering the population size enables standard error calculations.
Page 65: Coupons Case Study Conclusion (4/4)
Probability analysis emphasizes the likelihood of unusual redemption patterns leading to further investigations.