Lecture Notes: Sampling Distributions and Statistical Inference

Announcements

• Next course project posted on Canvas under the module section.
• Focus: One-sample T-test and QQ plot.
• Includes tasks like making histograms, calculating basic statistics.

Sampling Distributions

• Concept of Sampling Distribution:

  • Derived from taking multiple samples from a population and calculating the mean of each sample.
  • If you repeatedly collect samples and calculate their means, a distribution of these means forms (sampling distribution of the mean).
  • This distribution tends to be normal even if the population distribution is skewed.

• Example in R:

  • Creation of a skewed population.
  • Sampling of size 10 repeatedly to show the progression of the sampling distribution towards normality.
  • As sample size increases, the distribution of sample means becomes more normal.

Central Limit Theorem (CLT)

• Definition:

  • States that the sampling distribution of any statistic becomes normal as the sample size increases.

• Criteria for CLT to Hold:

  • Samples must be independent.
  • Observations should be randomly collected.

Importance of Sampling Distribution

• Mean of Sampling Distribution vs Population Mean:

  • The mean of the sampling distribution equals the population mean.
  • The sample mean is an unbiased estimate of the population mean.

• Standard Error:

  • Defined as the standard deviation of the sampling distribution.
  • Indicates natural fluctuations or error from sample to sample.

Standard Error and Confidence Intervals

• Standard Error (SE) Calculation:

  • Formula: SE = Population Standard Deviation / √Sample Size
  • In practice, use sample standard deviation since population standard deviation is often unknown.

• Confidence Interval (CI):

  • A range of values likely to contain the true population mean.
  • Calculated using the standard error.

• Implications of Sample Size:

  • Larger sample sizes reduce standard error and result in narrower confidence intervals (more precise estimates).

Challenges in Practical Research

• In real-world scenarios, researchers often have only one sample.
• True population standard deviation is usually unknown.
• Solution: Use sample standard deviation to estimate population standard deviation.

T-Distribution

• Introduction to T-Distribution:

  • Used when sample sizes are small (less than 1000).
  • Heavier tails than normal distribution.
  • Developed for estimating population parameters when sample sizes are small.

• Example Using T-Distribution:

  • Scenario: Researcher claims college students spend 9 hours a day on phones.
  • Small sample size collected, leads to calculation of sample mean and standard deviation.
  • Use T-distribution to estimate confidence interval and evaluate the claim.

• Historical Context:

  • Developed by William Gosset at Guinness Brewery under pseudonym "Student".
  • Addressed issues with small sample sizes using a normal distribution.

Summary

• Recap:

  • Importance of sampling distributions in statistical inference.
  • Use of central limit theorem to justify normality assumption.
  • Variability in sample estimates captured by standard error.
  • Confidence intervals provide a range for the likely population mean.

• Transition to T-Distribution:

  • Solution to problems with small sample sizes by using T-distribution as a model.