Hypothesis Testing

Definition and Process

Hypothesis testing is a statistical process used to evaluate claims about a population parameter based on sample data. It involves formulating hypotheses and determining whether to reject the null hypothesis based on statistical evidence.

• Null Hypothesis (H0): The default assumption or claim about the population parameter, often representing no effect or no difference.

• Alternative Hypothesis (H1): The claim that contradicts the null hypothesis, representing the presence of an effect or difference.

Types of Hypothesis Tests

The form of the hypothesis determines the type of test:

• Two-sided (Bilateral) Test: Tests for any difference (H0: θ = θ0, H1: θ ≠ θ0).

• Left-sided (Lower-tail) Test: Tests for a decrease (H0: θ ≥ θ0, H1: θ < θ0).

• Right-sided (Upper-tail) Test: Tests for an increase (H0: θ ≤ θ0, H1: θ > θ0).

Key Terms in Hypothesis Testing

• Test Statistic: A standardized value calculated from sample data, used to decide whether to reject H0. Examples include t, Z, χ2, and F statistics.

• Significance Level (α): The probability of rejecting H0 when it is true (Type I error). Common values are 0.01, 0.05, and 0.10.

Decision Rule Using p-value

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the observed value under H0. The decision rule is:

• If p-value < α, reject H0.

• If p-value ≥ α, do not reject H0.

Calculation of p-value depends on the test type:

• For two-sided tests:

• For left-sided tests:

• For right-sided tests:

• Where

One-Sample Mean Hypothesis Test

When testing the mean of a single population:

• If population variance is known or sample size is large, use Z-test:

• Test statistic:

• Critical values:

t-Distribution and Its Application

Origins and Properties

The t-distribution was developed by William Gosset ("Student") for small sample inference when population variance is unknown. It is symmetric and bell-shaped, but has heavier tails than the normal distribution.

t-Test for Small Samples

When population variance is unknown and sample size is small, use the t-test:

• Test statistic:

• Degrees of freedom:

Degrees of Freedom

Degrees of freedom represent the number of independent values in a sample that can vary. For a sample of size n, the degrees of freedom for estimating variance is n-1.

• Example: In a sample of 4, 5, 6, the mean is 5. If two values change, the last must adjust to keep the mean at 5.

t-Test for Unknown Variance

When population variance is unknown and sample size is small:

• Test statistic:

• Critical values:

Regression Analysis

Historical Context and Concept

Regression analysis studies the relationship between a dependent variable and one or more independent variables. It was first formalized by Galton in 1893, examining the relationship between parent and child heights.

• Dependent variable (Y): The outcome being predicted.

• Independent variable (X): The predictor or explanatory variable.

Regression Equation and Residuals

The regression equation models the expected value of Y given X:

• Estimated regression line:

• Residual:

Least Squares Method

The least squares method estimates the regression line by minimizing the sum of squared residuals:

• Normal equations for least squares:

Testing Regression Coefficients: t and F Tests

To test the significance of regression coefficients:

• t-test: Tests if the slope (β) is significantly different from zero.

• Test statistic: , where is the standard error of the slope.

• Critical values:

F-test: Compares explained variance to unexplained variance to test overall model significance.

• Test statistic:

• Critical value:

Regression Example: Advertising and Sales

Consider the relationship between advertising expenditure (X) and sales (Y):

Regression Output Interpretation

Key statistics from regression analysis:

• R-squared (결정계수): Proportion of variance in Y explained by X. Here, .

• Standard error: Measures the average distance between observed and predicted values.

• t-statistic and p-value: Used to test the significance of the slope.

• F-statistic: Tests overall model significance.

Example Interpretation: The regression model explains 82.91% of the variance in sales. The slope is statistically significant (p-value < 0.05), indicating a meaningful relationship between advertising and sales.

*Additional info: The notes and images cover hypothesis testing, t-distribution, and regression analysis, all directly relevant to college-level statistics topics, including formulas, decision rules, and practical examples.*