Statistical Inference: Comparing Two Populations and Linear Regression

Statistical Inference: Comparing Two Populations and Linear Regression

Overview

This unit focuses on the application of statistical inference methods to analyze single variables and the relationship between two variables. Students will learn to summarize data, perform hypothesis tests, construct confidence intervals, and model relationships using regression, both conceptually and with statistical software (RStudio).

Comparing Two Populations

Dependent (Paired) vs. Independent Samples

• Dependent (Paired) Samples: Two sets of observations are paired when each observation in one sample can be paired with an observation in the other sample (e.g., before-and-after measurements on the same subjects).

• Independent Samples: The samples are independent if the observations in one sample do not influence or relate to the observations in the other sample (e.g., comparing two different groups).

• Recognizing Sample Type: Correct identification is crucial for selecting the appropriate statistical test.

Hypothesis Testing for Means and Proportions

• Single Mean of Differences (Paired t-test): Used when comparing the means of paired samples.

• Difference Between Two Independent Means (Two-sample t-test): Used for comparing means from two independent groups.

• Difference Between Two Independent Proportions (Two-proportion z-test): Used for comparing proportions from two independent groups.

• Writing Hypotheses:

  • Null Hypothesis (H0): States no difference or effect (e.g., $\mu_1 - \mu_2 = 0$ or $p_1 - p_2 = 0$).

  • Alternative Hypothesis (Ha): States the expected difference or effect (e.g., $\mu_1 - \mu_2 \neq 0$).

Test Statistics and Confidence Intervals

• Test Statistic for Paired t-test:

  • $t = \frac{\bar{d} - \mu_0}{s_d / \sqrt{n}}$

  • Where $\bar{d}$ is the mean of the differences, $s_d$ is the standard deviation of the differences, $n$ is the number of pairs, and $\mu_0$ is the hypothesized mean difference (often 0).

• Test Statistic for Two-sample t-test (Equal Variances Not Assumed):

  • $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$

  • Where $\bar{x}_1, \bar{x}_2$ are sample means, $s_1^2, s_2^2$ are sample variances, $n_1, n_2$ are sample sizes.

• Test Statistic for Two-proportion z-test:

  • $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$

  • Where $\hat{p}_1, \hat{p}_2$ are sample proportions, $\hat{p}$ is the pooled proportion, $n_1, n_2$ are sample sizes.

• Confidence Intervals:

  • For paired differences: $\bar{d} \pm t^* \frac{s_d}{\sqrt{n}}$

  • For two means: $(\bar{x}_1 - \bar{x}_2) \pm t^* \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}$

  • For two proportions: $(\hat{p}_1 - \hat{p}_2) \pm z^* \sqrt{\frac{\hat{p}_1(1-\hat{p}_1)}{n_1} + \frac{\hat{p}_2(1-\hat{p}_2)}{n_2}}$

Interpreting Results

• Conclusion: State whether the evidence supports the alternative hypothesis, referencing the context of the data and the p-value.

• p-value: The probability of observing a test statistic as extreme as, or more extreme than, the observed value under the null hypothesis.

• Statistical Significance: If the p-value is less than the significance level (e.g., 0.05), reject the null hypothesis.

Linear Regression Analysis

Least Squares Regression Line

• Purpose: To model the relationship between a numerical response variable and a single explanatory variable.

• Equation: $\hat{y} = b_0 + b_1 x$

• Interpretation:

  • Slope ($b_1$): The estimated change in the response variable for a one-unit increase in the explanatory variable.

  • Intercept ($b_0$): The estimated value of the response variable when the explanatory variable is zero (may not always be meaningful in context).

• Prediction: Use the regression equation to predict the response for a given value of the explanatory variable.

Residuals

• Definition: The difference between the observed value and the value predicted by the regression line.

• Formula: $e_i = y_i - \hat{y}_i$

• Interpretation: Residuals help assess the fit of the regression model.

Coefficient of Determination ($R^2$)

• Definition: The proportion of the variance in the response variable that is explained by the explanatory variable.

• Formula: $R^2 = 1 - \frac{SS_{res}}{SS_{tot}}$

• Interpretation: An $R^2$ value close to 1 indicates a strong linear relationship; close to 0 indicates a weak relationship.

Using RStudio for Statistical Analysis

• prop.test: Used to test hypotheses about proportions and to construct confidence intervals for proportions.

• t.test: Used for testing means (one-sample, paired, or two-sample) and constructing confidence intervals for means.

• Calculating p-values: RStudio functions return exact p-values for hypothesis tests.

• Regression Analysis: RStudio can fit least squares regression lines and calculate $R^2$.

Critical Evaluation of Data-Based Claims

• Interpretation in Context: Conclusions should be stated in plain language, avoiding unnecessary statistical jargon.

• Critique: Evaluate the appropriateness of the statistical methods used and the validity of the conclusions drawn from data analyses.

Summary Table: Key Tests and When to Use Them

Test	When to Use	Key Formula
Paired t-test	Comparing means of dependent (paired) samples	$t = \frac{\bar{d} - \mu_0}{s_d / \sqrt{n}}$
Two-sample t-test	Comparing means of two independent samples	$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
Two-proportion z-test	Comparing proportions of two independent samples	$z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$
Linear Regression	Modeling relationship between two numerical variables	$\hat{y} = b_0 + b_1 x$

Example: Paired t-test

• Suppose a group of students takes a test before and after a training session. The paired t-test can be used to determine if the training had a significant effect on scores.

Example: Linear Regression

• Predicting a student's final exam score based on the number of hours studied using the regression equation.

Additional info: Some formulas and context were inferred to provide a complete, self-contained study guide.