Unit 5: Theory of Estimation and Testing of Hypothesis
Course: Statistics for Economics (ECODSC 253)
This final unit acts as the culmination of the course, providing the framework for statistical inference where we use sample data to make generalized statements about whole populations.
1. Theory of Estimation: Point vs. Interval
Estimation involves using a sample statistic to predict a population parameter.
- Point Estimation: Provides a single numerical value as the estimate of the parameter. For example, using the sample mean as the single best guess for the population mean.
- Interval Estimation: Provides a range of values (confidence interval) within which the parameter is expected to lie with a certain degree of confidence.
2. Characteristics of a Good Estimator
A "good" estimator must possess certain mathematical properties to be reliable:
- Unbiasedness: The expected value of the estimator should equal the true population parameter.
- Consistency: As the sample size increases, the estimator should get closer and closer to the true population parameter.
- Efficiency: Among all unbiased estimators, the one with the smallest variance is the most efficient.
- Sufficiency: The estimator uses all the information about the parameter contained in the sample.
3. Concepts of Testing Hypothesis & Significance
Hypothesis testing is a formal procedure for investigating ideas about the world using statistics.
- Null Hypothesis (H0): A statement of "no effect" or "no difference" that we seek to test against.
- Alternative Hypothesis (H1): The statement that we suspect is actually true and is the opposite of the null hypothesis.
- Level of Significance (alpha): The probability of rejecting the null hypothesis when it is actually true. Common levels are 5% or 1%.
4. Type I and Type II Errors
Because we use samples, there is always a chance of reaching the wrong conclusion.
5. Statistical Tests: Z, t, Chi-Square, and F Distributions
The choice of test depends on the sample size and what is being compared.
Large Sample Tests (Z-test)
Used when the sample size is large (n > 30). It follows the standard normal distribution.
Small Sample Tests (t-test)
Used when the sample size is small (n < 30) and the population standard deviation is unknown.
[Image comparing the Z-distribution bell curve with the flatter t-distribution bell curve]
Chi-Square Test (χ²)
Primarily used for testing the "Goodness of Fit" (how well observed data matches expected data) or the independence of attributes in a contingency table.
F-Test
Used to compare the variances of two different populations. It is the foundation for Analysis of Variance (ANOVA).
Exam Tip: Which Test to Choose?
- If n > 30 and you know the variance, always use the Z-test.
- If n < 30 and variance is unknown, switch to the t-test.
- Use Chi-Square when dealing with frequencies or categories rather than means.
- Use F-test only when the question explicitly asks to compare "variability" or "variances."