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1. Population and Sample

• Population: Entire group under study (finite, infinite, real, hypothetical).

• Sample: Subset of the population used for analysis.

  • Sample Size: Number of elements in the sample.

  • Sampling Methods: Random, stratified, etc., to ensure representativeness.

2. Sampling Distribution

• Distribution of a statistic (e.g., mean) from multiple samples.

• Key Concepts:

  • Mean of Sampling Distribution: Equals population mean (\mu_{\bar{X}} = \mu).

  • Standard Error (SE): \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} (decreases as n increases).

  • Central Limit Theorem (CLT): For n \geq 30, sampling distribution of \bar{X} is approximately normal, regardless of population shape.

3. Estimation

• Point Estimate: Single value (e.g., \bar{x} for \mu, s for \sigma).

• Interval Estimate: Range with confidence level (e.g., 95% CI).

  • Margin of Error: Z_{\alpha/2} \times \text{SE} .

  • CI for Mean:

  • \sigma known: \bar{x} \pm Z_{\alpha/2} \frac{\sigma}{\sqrt{n}}.

  • \sigma unknown: Use t-distribution (for small samples: n < 30).

4. Regression and Correlation

• Regression: Models relationship between dependent (Y) and independent (X) variables.

  • Equation: Y = a + bX .

  • Slope (b): b = \frac{\sum (xi - \bar{x})(yi - \bar{y})}{\sum (x_i - \bar{x})^2} .

  • Intercept (a): a = \bar{y} - b\bar{x} .

• Correlation (Pearson’s r): Measures linear relationship strength (-1 \leq r \leq 1).

  • r = 1: Perfect positive; r = -1: Perfect negative; r = 0: No correlation.

5. Time Series Analysis

• Definition: Data collected at regular intervals to identify trends.

• Components:

  1. Secular Trend (long-term direction).

  2. Seasonal Variation (periodic fluctuations).

  3. Cyclical Variations (economic cycles).

  4. Irregular Variation (random noise).

• Methods:

  • Least Squares: Fit trend line y = a + bt .

  • Moving Average: Smooths data to highlight trends.

Key Formulas

• Standard Error (Mean): \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} .

• Correlation Coefficient:
  $$ r = \frac{n\sum xy - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum