Methods for Quantitative Response Variables – One and Two Groups

Methods for Quantitative Response Variables – One and Two Groups

Overview

This chapter explores statistical methods for analyzing quantitative response variables when comparing one or two groups. It covers hypothesis testing and confidence intervals for means and variances, including one-sample and two-sample scenarios, as well as paired samples. The appropriate test depends on the study design and the assumptions about the data.

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 from a known or hypothesized population mean when the population standard deviation is unknown.

• Null hypothesis (H0): $\mu = \mu_0$

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

Assumptions:

• Random sample from the population of interest

• Observations are independent

• Population is approximately normal, or sample size is large (Central Limit Theorem)

Test Statistic:

$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$

where $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, and $n$ is the sample size.

Confidence Interval for the Mean:

$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$

where $t^*$ is the critical value from the t-distribution with $n-1$ degrees of freedom.

Example: Leaf Nitrogen Content

A botanist measures the nitrogen content in leaves from a wildflower species. The sample mean is 3.41% with a standard deviation of 0.15% (n = 15). Test if the mean differs from 3.1% at the 5% significance level.

Hypotheses: $H_0: \mu = 3.1$, $H_A: \mu \neq 3.1$

Test statistic: $t = \frac{3.41 - 3.1}{0.15/\sqrt{15}} = 7.98$

With 14 degrees of freedom, compare to the critical value or calculate the p-value.

Chi-Square Test for a Variance

This test evaluates whether the population variance equals a specified value $\sigma_0^2$.

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

• Alternative hypothesis (HA): $\sigma^2 \neq \sigma_0^2$, $\sigma^2 > \sigma_0^2$, or $\sigma^2 < \sigma_0^2$

Test Statistic:

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

where $s^2$ is the sample variance.

Confidence Interval for Variance:

$\left( \frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}, \frac{(n-1)s^2}{\chi^2_{\alpha/2}} \right)$

Example: Manufacturing Variability

A quality control engineer tests if the variance in spring length is 0.04. With $n = 24$, $s^2 = 0.06$, the test statistic is $\chi^2 = \frac{23 \times 0.06}{0.04} = 34.5$.

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): $\mu_1 = \mu_2$

• Alternative hypothesis (HA): $\mu_1 \neq \mu_2$, $\mu_1 > \mu_2$, or $\mu_1 < \mu_2$

Pooled Estimate of Variance:

$s_p^2 = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1 + n_2 - 2}$

Test Statistic:

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

Assumptions: Independent random samples, populations are normally distributed, equal variances.

Welch’s (Nonpooled) t-Test

Used when the assumption of equal variances is not valid. This test adjusts the standard error and degrees of freedom.

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 are calculated using Welch’s formula.

Example: Ad Campaign Effectiveness

Comparing mean sales increases from two advertising campaigns using the appropriate t-test based on variance equality.

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 test if they are equal.

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

• Alternative hypothesis (HA): $\sigma_1^2 \neq \sigma_2^2$

Test Statistic:

$F = \frac{s_1^2}{s_2^2}$

Assumptions: Independent random samples, normality in both populations.

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: Enzyme Activity Variability

Testing if the variability in enzyme activity differs between two species using the F-test.

12.4 Paired Samples: Matched-Pairs t-Test

Matched-Pairs t-Test

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

• Null hypothesis (H0): $\mu_d = 0$ (no difference in means)

• Alternative hypothesis (HA): $\mu_d \neq 0$, $\mu_d > 0$, or $\mu_d < 0$

Test Statistic:

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

where $\bar{d}$ is the mean of the differences, $s_d$ is the standard deviation of the differences, and $n$ is the number of pairs.

Assumptions: Differences are approximately normally distributed, pairs are independent.

Example: Teaching Methods

Comparing reading test scores for students taught by two methods using matched pairs. The test evaluates whether the new method leads to higher scores than the standard method.

Summary Table of Key Tests and Formulas

Test	Formula	Purpose
One-sample t-test	$t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$	Test mean of one group
Confidence interval for mean	$\bar{x} \pm t^* \frac{s}{\sqrt{n}}$	Estimate population mean
Chi-square test for variance	$\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}$	Test variance of one group
Pooled t-test	$t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}}$	Compare means (equal variances)
Welch’s t-test	$t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$	Compare means (unequal variances)
F-test	$F = \frac{s_1^2}{s_2^2}$	Compare variances
Matched-pairs t-test	$t = \frac{\bar{d}}{s_d/\sqrt{n}}$	Compare paired means

Key Points

• Choose the appropriate test based on the number of groups, independence of samples, and equality of variances.

• Check assumptions before applying each test.

• Interpret results in the context of the research question.