4.3 One-Sample and Two-Sample Tests

One-sample and two-sample tests are key tools in statistical inference. They help us compare sample data to population parameters or between two groups. These tests build on the foundation of confidence intervals and hypothesis testing, allowing us to make decisions about populations based on sample evidence.

Understanding these tests is crucial for drawing valid conclusions from data. We'll explore how to conduct and interpret various one-sample and two-sample tests, including t-tests, z-tests, and proportion tests. We'll also learn when to use each test and how to choose the right one for different situations.

One-sample hypothesis testing

Fundamentals of one-sample tests

Top images from around the web for Fundamentals of one-sample tests

• 

  Hypothesis Testing: One Sample | Boundless Statistics View original

  Is this image relevant?

• 

  Hypothesis Testing (4 of 5) | Concepts in Statistics View original

  Is this image relevant?

• 

  Comparing two means – Learning Statistics with R View original

  Is this image relevant?

• 

  Hypothesis Testing: One Sample | Boundless Statistics View original

  Is this image relevant?

• 

  Hypothesis Testing (4 of 5) | Concepts in Statistics View original

  Is this image relevant?

1 of 3

Top images from around the web for Fundamentals of one-sample tests

• 

  Hypothesis Testing: One Sample | Boundless Statistics View original

  Is this image relevant?

• 

  Hypothesis Testing (4 of 5) | Concepts in Statistics View original

  Is this image relevant?

• 

  Comparing two means – Learning Statistics with R View original

  Is this image relevant?

• 

  Hypothesis Testing: One Sample | Boundless Statistics View original

  Is this image relevant?

• 

  Hypothesis Testing (4 of 5) | Concepts in Statistics View original

  Is this image relevant?

1 of 3

• Compare single sample statistic to known or hypothesized population parameter
• One-sample t-test compares sample mean to hypothesized population mean when population standard deviation unknown
• One-sample z-test used when population standard deviation known or large sample sizes (n > 30)
• One-sample proportion tests compare sample proportion to hypothesized population proportion
• Critical value approach and p-value approach draw conclusions in hypothesis testing
• Assumptions involve random sampling, independence of observations, normality (t-tests) or normal approximation conditions (proportion tests)
• Effect size measures (Cohen's d) assess practical significance beyond statistical significance

Conducting one-sample tests

• Calculate test statistic using appropriate formula based on test type (t-test, z-test, proportion test)
• For t-test: $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$ where $\bar{x}$ is sample mean, $\mu_0$ is hypothesized population mean, s is sample standard deviation, n is sample size
• For z-test: $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$ where $\sigma$ is known population standard deviation
• For proportion test: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ where $\hat{p}$ is sample proportion, $p_0$ is hypothesized population proportion
• Determine degrees of freedom (df = n - 1 for t-tests)
• Compare test statistic to critical value or calculate p-value
• Make decision to reject or fail to reject null hypothesis based on significance level (α)

Interpreting one-sample test results

• Interpret p-value meaning probability of obtaining test statistic as extreme as observed, assuming null hypothesis true
• Smaller p-values provide stronger evidence against null hypothesis
• Consider practical significance alongside statistical significance
• Calculate confidence intervals to estimate population parameter (mean or proportion)
• Report effect sizes (Cohen's d for means, h for proportions) to quantify magnitude of difference
• Contextualize results within research question and real-world implications
• Acknowledge limitations (sample size, assumptions) when drawing conclusions

Two-sample hypothesis testing

Fundamentals of two-sample tests

• Compare parameters (means or proportions) between two independent populations using sample data
• Independent samples t-test compares means between unrelated groups with unknown, assumed equal population standard deviations
• Welch's t-test adapts independent samples t-test for assumed unequal population variances
• Two-sample z-test for means used with known population standard deviations or large sample sizes
• Two-sample test for proportions compares proportions between independent populations
• Levene's test assesses equality of variances assumption for t-tests
• Assumptions include independent random samples, independence within/between groups, normality (t-tests) or normal approximation conditions (proportion tests)

Conducting two-sample tests

• Calculate test statistic using appropriate formula based on test type and assumptions
• For independent samples t-test: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_p^2}{n_1} + \frac{s_p^2}{n_2}}}$ where $s_p^2$ is pooled variance
• For Welch's t-test: $t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}$ where $s_1^2$ and $s_2^2$ are sample variances
• For two-sample z-test: $z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$ where $\sigma_1^2$ and $\sigma_2^2$ are known population variances
• For two-sample proportion test: $z = \frac{\hat{p}_1 - \hat{p}_2}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_1} + \frac{1}{n_2})}}$ where $\hat{p}$ is pooled sample proportion
• Determine degrees of freedom (varies based on test type and sample sizes)
• Compare test statistic to critical value or calculate p-value
• Make decision to reject or fail to reject null hypothesis based on significance level (α)

Interpreting two-sample test results

• Interpret p-value meaning probability of obtaining difference as extreme as observed, assuming null hypothesis true
• Smaller p-values provide stronger evidence against null hypothesis of no difference between populations
• Calculate and interpret confidence intervals for difference between population parameters
• Report effect sizes (Cohen's d for means, h for proportions) to quantify magnitude of difference between groups
• Consider practical significance of observed differences in context of research question
• Acknowledge limitations (sample sizes, assumptions) when generalizing results to populations
• Discuss potential sources of between-group differences and implications for further research

Paired t-tests for related samples

Fundamentals of paired t-tests

• Compare means between two related groups or repeated measurements on same subjects
• Based on differences between paired observations, reducing problem to one-sample test on differences
• Increase statistical power by reducing variability associated with individual differences
• Replace assumption of independence between pairs with assumption of independence of differences
• Other assumptions include normality of differences and absence of significant outliers in differences
• Calculate effect size using Cohen's d for paired samples
• Common in before-after studies (weight loss program), matched-pairs designs (twins), and repeated measures experiments (drug effectiveness over time)

Conducting paired t-tests

• Calculate differences between paired observations (d = x2 - x1)
• Compute mean difference ($\bar{d}$) and standard deviation of differences (sd)
• Calculate test statistic: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$ where n is number of pairs
• Determine degrees of freedom (df = n - 1)
• Compare test statistic to critical value or calculate p-value
• Make decision to reject or fail to reject null hypothesis based on significance level (α)
• Calculate confidence interval for mean difference: $\bar{d} \pm t_{\alpha/2} \frac{s_d}{\sqrt{n}}$

Interpreting paired t-test results

• Interpret p-value meaning probability of obtaining difference as extreme as observed, assuming no true difference
• Smaller p-values provide stronger evidence against null hypothesis of no difference between paired measurements
• Consider magnitude and direction of mean difference in context of research question
• Calculate and interpret effect size (Cohen's d for paired samples) to quantify practical significance
• Discuss implications of results for research hypothesis and real-world applications
• Acknowledge limitations (sample size, potential confounds) when generalizing results
• Compare advantages of paired design to independent samples approach for specific research context

Choosing the right hypothesis test

Factors influencing test selection

• Research question determines primary focus (comparing means, proportions, or relationships)
• Number of groups being compared guides choice between one-sample, two-sample, or multi-group tests
• Nature of data (continuous or categorical) influences selection of parametric or non-parametric tests
• Sample size and knowledge of population parameters inform decision between z-tests and t-tests
• Level of measurement (nominal, ordinal, interval, or ratio) of dependent variable directs choice of test
• Assumption of independence between observations determines appropriateness of paired or independent samples test
• Sampling method and study design crucial in selecting correct statistical test (random sampling, experimental vs. observational)

Preliminary considerations and tests

• Assess normality of data using visual methods (Q-Q plots, histograms) or statistical tests (Shapiro-Wilk, Kolmogorov-Smirnov)
• Check for outliers using boxplots or z-scores to identify potential influential observations
• Evaluate homogeneity of variances using Levene's test for independent samples t-tests
• Consider robustness of different tests to violations of assumptions when choosing between options
• Examine sample sizes to determine appropriateness of large-sample approximations or need for exact tests
• Assess independence of observations through study design and data collection methods
• Consider power analysis to determine if sample size sufficient to detect meaningful effects

Decision-making process for test selection

• Identify research question and hypothesis (difference between groups, relationship between variables)
• Determine number of groups or variables involved (one-sample, two-sample, multi-group, correlation)
• Classify variables as independent (predictor) or dependent (outcome) measures
• Assess level of measurement for each variable (nominal, ordinal, interval, ratio)
• Evaluate whether data meet assumptions for parametric tests (normality, homogeneity of variances)
• Consider alternatives if assumptions violated (non-parametric tests, data transformations)
• Assess independence of observations or need for paired design
• Consult decision trees or flowcharts to guide selection process based on above factors
• Seek expert advice or statistical consultation for complex designs or uncertainty in test selection