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.