Methods for Quantitative Response Variables – One and Two Groups: Statistical Tests and Applications

Methods for Quantitative Response Variables – One and Two Groups

Overview

This section covers statistical methods for analyzing quantitative response variables in one and two group designs. The focus is on hypothesis testing and confidence intervals for means and variances, including one-sample and two-sample tests, as well as paired sample analysis.

12.1 One-Sample Tests for the Mean and Variance

One-Sample t-Test for a Mean

The one-sample t-test is used to determine whether the mean of a single sample differs significantly from a hypothesized population mean.

• Null hypothesis (H0): $\mu = \mu_0$

• Alternative hypothesis (HA): $\mu \neq \mu_0$ (two-sided) or $\mu > \mu_0$, $\mu < \mu_0$ (one-sided)

• Assumptions:

  • Random sample from the population

  • Observations are independent

  • Population is approximately normally distributed (especially important for small samples)

• Test statistic: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$

• Distribution: Student's t-distribution with $n-1$ degrees of freedom

• Confidence interval for mean: $\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$

Example: Testing leaf nitrogen content in a sample of wildflower species.

Chi-Square Test for a Variance

Used to test hypotheses about the population variance or to construct confidence intervals for the variance.

• Null hypothesis (H0): $\sigma^2 = \sigma_0^2$

• Test statistic: $\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$

• Distribution: Chi-square distribution with $n-1$ degrees of freedom

• Confidence interval for variance: $\left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}} \right)$

Example: Assessing manufacturing variability in spring length.

12.2 Two Independent Samples: Student's t-Test and Welch's t-Test

Student's (Pooled) t-Test

Used to compare means from two independent samples when population variances are assumed equal.

• Null hypothesis (H0): $\mu_1 = \mu_2$

• Pooled variance estimate: $s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}$

• Test statistic: $t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$

• Distribution: t-distribution with $n_1 + n_2 - 2$ degrees of freedom

• Assumptions:

  • Independent random samples

  • Populations are normally distributed

  • Equal population variances

Welch's (Nonpooled) t-Test

Used when population variances are not assumed equal.

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

• Degrees of freedom: Approximated using Welch's formula

• Assumptions:

  • Independent random samples

  • Populations are normally distributed

  • Unequal population variances

Example: Comparing ad campaign effectiveness between two groups.

12.3 Equality of Variances: F-Test and Alternatives

F-Test for Equality of Variances

The F-test compares the variances of two independent samples to evaluate if they are significantly different.

• Null hypothesis (H0): $\sigma_1^2 = \sigma_2^2$

• Test statistic: $F = \frac{s_1^2}{s_2^2}$

• Distribution: F-distribution with $n_1-1$ and $n_2-1$ degrees of freedom

• Assumptions:

  • Independent random samples

  • Each population is normally distributed

• Confidence interval for ratio of variances: $\left( \frac{s_1^2}{s_2^2} \cdot \frac{1}{F_{1-\alpha/2}}, \frac{s_1^2}{s_2^2} \cdot \frac{1}{F_{\alpha/2}} \right)$

Example: Testing enzyme activity variability between two species.

12.4 Paired Samples: Matched-Pairs t-Test

Matched-Pairs t-Test

Used when each subject provides two related measurements (e.g., before and after treatment, or matched pairs). The test focuses on the mean difference between paired observations.

• Null hypothesis (H0): $\mu_D = 0$ (mean difference is zero)

• Test statistic: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$

• Distribution: t-distribution with $n-1$ degrees of freedom

• Assumptions:

  • Differences are paired and independent

  • Differences are approximately normally distributed

Example: Comparing reading test scores between two teaching methods using matched pairs.

Recap: Key Formulas and Concepts

Keyword/Concept	Definition/Formula
One-sample t-test	$t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$
Confidence interval for mean	$\bar{x} \pm t_{\alpha/2, n-1} \frac{s}{\sqrt{n}}$
Chi-square test for variance	$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$
Confidence interval for variance	$\left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2, n-1}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2, n-1}} \right)$
Pooled t-test	$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$
Welch's (nonpooled) t-test	$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$
F-test	$F = \frac{s_1^2}{s_2^2}$
Confidence interval for variance ratio	$\left( \frac{s_1^2}{s_2^2} \cdot \frac{1}{F_{1-\alpha/2}}, \frac{s_1^2}{s_2^2} \cdot \frac{1}{F_{\alpha/2}} \right)$
Paired t-test	$t = \frac{\bar{d}}{s_d / \sqrt{n}}$

Additional info:

• Examples and step-by-step procedures for performing tests in JMP statistical software are included in the original notes.

• Decision trees and flowcharts are used to guide test selection based on data structure and assumptions.