Key Concepts in Sampling and Statistics

Oct 2, 2024

Lecture Notes: Sampling and Statistical Concepts

Sampling and Statistics

  • Continuation of sampling chapter; discussed finite population correction factor (FPC).
  • FPC used when dealing with finite populations; equation: ( \sqrt{\frac{N-n}{N-1}} ).
  • Utilized when ( \frac{n}{N} \geq 0.05 ).
  • Originates from hypergeometric distribution.

Key Statistical Theorems

  • Central Limit Theorem (CLT): For large samples, the sampling distribution of the sample mean is approximately normal.
  • Markov and Chebyshev Inequalities: Applied in theoretical statistics.

Hypergeometric Distribution

  • Discrete probability distribution for ( k ) successes in ( n ) draws without replacement.

Sampling Distributions

  • Normal distribution approximation used in large populations.
  • Population parameters: ( \sigma ) (population standard deviation) vs. ( s ) (sample standard deviation).
  • Normal Distribution: Assumed for large sample sizes (n ( \geq ) 30).

Example: Coin Toss Experiment

  • Example with 100 coin tosses; demonstrated random sampling and observed normal distribution.

Proportions and Sampling

  • Proportions (%) are key in sampling, tend to be normally distributed.
  • Finite Population Correction: Applied in proportion sampling.

Interval Estimation

  • General Formula: ( \hat{\theta} \pm M(x) ), where ( M ) is the margin of error.
  • Used for estimating population parameters.
  • Confidence Intervals: Express uncertainty in parameter estimation.

T-Distribution

  • Used when population standard deviation ( \sigma ) is unknown.
  • Approximates normal distribution as degrees of freedom increase.

Confidence Intervals and Significance

  • Confidence Level: 1 - ( \alpha ); probability that interval contains population mean.
  • Level of Significance (( \alpha )): Probability that interval does not contain population mean.
  • Confidence intervals centered around sample mean and defined by margin of error.

Example Problems and Applications

  • Credit card balances and estimation using sample data.
  • Discussed application in Excel and R for calculations.

Upcoming Topics

  • Next lecture to cover hypothesis tests.

These notes summarize the key points from the lecture on sampling and statistical methods, focusing on the finite population correction factor, sampling distributions, interval estimation, and confidence intervals. The notes also provide an overview of upcoming topics related to hypothesis testing.