Lecture on Behavior of Sample Proportions

Main Focus

• Objective: Investigate how sample size impacts the spread and sample proportions.
• Context: Analysis based on part-time college students, where 60% are assumed to be female.

Investigating Sample Size Impact

• Sample Distributions:
  • Two distributions examined:
    • Sample size of 25 (previously studied)
    • Sample size of 100

Previous Observations (Sample Size: 25)

• Normal Distribution:
  • Mean (p-hat values): 0.6
  • Standard Deviation: 0.1
  • Typical values fell between 0.5 and 0.7

Current Investigation (Sample Size: 100)

• Expectation:
  • Query whether an increase in sample size affects variability:
    • More variability?
    • Less variability?
    • Same variability?
• Experiment Setup:
  • 100 individuals sampled randomly
  • Initial sample example: 67 females from 100 students (p-hat = 0.67)

Results

• Observations:
  • Less variability observed in sample proportions for sample size 100 compared to 25.
  • Standard deviation decreased when sample size increased from 25 to 100:
    • Sample size 25: Standard deviation ~0.1
    • Sample size 100: Standard deviation ~0.05
• Conclusion:
  • Larger samples result in less variability.
  • Mean of sampling distribution remains unchanged at 0.6.
  • Distribution observed to be normal.

Future Exploration

• Text Discussion:
  • Advanced probability theory will be used to calculate the standard deviation of sample proportions.
  • Explore conditions where the distribution might not be normal.