One-sample and two-sample tests are key tools in hypothesis testing. They help us compare data to known values or between groups. These tests, like t-tests and z-tests, let us draw conclusions about populations based on sample data.

Choosing the right test depends on your research question and data type. One-sample tests compare a single group to a known value, while two-sample tests compare two groups. Understanding when to use each test is crucial for accurate statistical analysis.

One-sample vs Two-sample Tests

Distinguishing Between Test Types

• One-sample tests compare single sample mean or proportion to known population parameter or hypothesized value
• Two-sample tests compare means or proportions between two independent groups or paired observations
• Selection depends on research question, study design, and available data
• One-sample tests suit before-after studies, quality control assessments, comparisons to established standards
• Two-sample tests fit comparing outcomes between treatment and control groups, examining differences between distinct populations
• Data nature (continuous or categorical) and sample size influence specific test selection (t-test, z-test, non-parametric alternatives)
• Null and alternative hypotheses formulation determines one-sample or two-sample test requirement

Applications and Examples

• One-sample test examines whether average test scores in a class (sample) differ from the school's historical average (population parameter)
• Two-sample independent test compares mean blood pressure between patients receiving new drug vs placebo
• Two-sample paired test evaluates weight loss in individuals before and after a diet program
• One-sample proportion test assesses if the proportion of defective items in a production batch exceeds the acceptable limit
• Two-sample proportion test compares voter turnout rates between two different electoral districts

Performing and Interpreting t-tests

One-sample t-tests

• Compare sample mean to known or hypothesized population mean, assuming normality
• Calculate t-statistic by dividing difference between sample and population means by standard error
• Degrees of freedom equal sample size minus one (n - 1)
• Formula: $t = \frac{\bar{x} - \mu}{s / \sqrt{n}}$
  • $\bar{x}$ sample mean
  • $\mu$ population mean
  • $s$ sample standard deviation
  • $n$ sample size
• Example: Testing if average height of basketball players in a league differs from national average

Two-sample t-tests

• Independent samples t-test compares means between two unrelated groups
• Dependent samples t-test (paired t-test) compares means of paired observations
• Independent t-test calculates t-statistic using difference between two sample means divided by standard error of the difference
• Levene's test checks equal variances assumption, determines use of pooled or separate variance estimates
• Degrees of freedom calculation varies based on equal or unequal variance assumption
• Paired t-test uses differences between paired observations, degrees of freedom equal number of pairs minus one
• Example: Comparing average salaries between two different industries (independent) or employee satisfaction before and after training program (paired)

Interpreting t-test Results

• Examine t-statistic, degrees of freedom, p-value, confidence intervals to draw conclusions about null hypothesis
• Calculate and report effect size measures (Cohen's d) to indicate observed difference magnitude
• Cohen's d formula: $d = \frac{\bar{x}_1 - \bar{x}_2}{s_{\text{pooled}}}$
• Interpret effect sizes: small (0.2), medium (0.5), large (0.8)
• Example interpretation: "The t-test revealed a significant difference (t(58) = 3.24, p < .01, d = 0.85) between treatment and control groups, with a large effect size"

Z-tests for Proportions

One-sample Proportion Tests

• Compare sample proportion to known or hypothesized population proportion
• Calculate z-statistic using difference between observed and expected proportions, divided by standard error of proportion
• Formula: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$
  • $\hat{p}$ sample proportion
  • $p_0$ hypothesized population proportion
  • $n$ sample size
• Require sufficiently large sample sizes (np ≥ 5 and n(1-p) ≥ 5) for normal approximation to binomial distribution
• Example: Testing if proportion of left-handed people in a sample differs from known population proportion

Two-sample Proportion Tests

• Compare proportions between two independent groups
• Use pooled estimate of proportion to calculate standard error under null hypothesis of no difference
• Formula: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$
  • $\hat{p}_1, \hat{p}_2$ sample proportions
  • $\hat{p}$ pooled proportion estimate
  • $n_1, n_2$ sample sizes
• Apply continuity correction (Yates' correction) to improve approximation of discrete binomial by continuous normal distribution
• Example: Comparing proportion of smokers between two different age groups or geographic regions

Interpreting Z-test Results

• Examine z-statistic, p-value, confidence intervals to draw conclusions about null hypothesis
• Calculate confidence intervals for proportions using formula: $\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$
• Interpret results in context of research question and practical significance
• Example interpretation: "The z-test showed a significant difference in proportions (z = 2.86, p < .01), with a 95% confidence interval for the difference in proportions of [0.03, 0.15]"

Choosing the Right Test Statistic

Factors Influencing Test Selection

• Data type (continuous or categorical) guides choice between t-tests and z-tests for proportions
• Sample size determines use of t-statistics (small samples) or z-statistics (large samples, n > 30)
• Population standard deviation knowledge influences z-statistic use (known σ) vs t-statistic (unknown σ)
• Null and alternative hypotheses determine one-tailed or two-tailed tests, affecting p-value calculation and interpretation
• Assumptions about population distribution shape test statistic choice (normal distribution for parametric tests)
• Example: Choose z-test for large sample (n = 500) comparing proportion of college graduates in two cities, t-test for small sample (n = 20) comparing mean exam scores between two classes

Calculating and Interpreting P-values

• P-values represent probability of obtaining test statistic as extreme as or more extreme than observed value, assuming null hypothesis true
• Calculate t-test p-values using t-distribution with appropriate degrees of freedom
• Determine z-test p-values using standard normal distribution
• Compare significance level (α) to p-value for hypothesis testing decision-making
• Use critical values from statistical tables or software as alternative method for decision-making
• Example interpretation: "The calculated p-value of 0.003 is less than the significance level of 0.05, leading to rejection of the null hypothesis"

Practical Considerations

• Consider practical significance alongside statistical significance when interpreting results
• Report effect sizes and confidence intervals to provide comprehensive understanding of findings
• Account for multiple comparisons when conducting several tests to control overall Type I error rate (Bonferroni correction)
• Evaluate test assumptions and consider non-parametric alternatives when assumptions violated
• Example: Wilcoxon rank-sum test as non-parametric alternative to independent samples t-test when normality assumption violated