Central Limit Theorem Lecture Notes

Introduction

• Discussion of the Central Limit Theorem (CLT)
• Usage of StatCrunch application for demonstration

Understanding the Population and Sampling

• Population Assumption: We assume an underlying truth about the population.
• Sampling: Realistically, we can take samples from the population.
  • Example: Taking a sample of size 2.
  • Calculated Sample Mean: 24
  • Standard Deviation between samples: 25
  • Mean of Sample Means (( \mu_{\bar{x}} )): 24.24
  • Standard Deviation of Sample Mean (( \sigma_{\bar{x}} )) is calculated.
• Observation: More samples result in data resembling the population.

Importance of Large Sample Sizes

• Taking 1,000 samples of sample size 2:
  • Mean of sample means aligns closely with population mean.
  • Individual sample means vary.
• Formula: ( \sigma_{\bar{x}} = \sigma / \sqrt{n} )
  • ( \sigma = 5 ), sample size ( n = 2 )
  • ( \sigma_{\bar{x}}^2 = 12.5 ), ( \sigma_{\bar{x}} \approx 3.5355 )_

Central Limit Theorem in Practice

• Theoretical expectation: Larger sample sizes yield a normal bell-shaped distribution.
• Uniform Distribution Example:
  • Population with normal distribution mean = 24, standard deviation = 7.5.
  • Range assumed: 11 to 35.
  • With 1,000 samples, the distribution of sample means becomes bell-shaped.
• Sample Size of 30:
  • With sample size 30, single sampling efforts start looking normal.
  • Multiple sampling continues to approach normal distribution.

Mathematical Validation

• Mean of Sample Means: Should equal the population mean, very close in results.
• Standard Deviation of Sample Means:
  • Calculated: ( 7.5^2 / 30 ) and its square root.
  • Theory: ( \approx 1.369 ), observed: ( 1.382 ).

Conclusion

• The CLT demonstrates that larger sample sizes will have a distribution that mirrors a normal distribution.
• This theorem forms the basis of inferential statistics.

Further Support

• Encouragement to bring questions to class for further clarification.
• Acknowledgment of the theoretical nature of the topic and willingness to provide additional demonstrations.