Glossary

9.2 Paired Samples T-Test

2 min read•july 23, 2024

The compares two related measurements, like before-after scenarios or . It's used to determine if there's a significant difference between paired observations, making it perfect for pre-post studies or comparing methods on the same subjects.

To conduct the test, you calculate differences between pairs, compute mean and standard deviation, and use a formula to find the . Comparing this to critical values helps decide if there's a significant difference. Confidence intervals provide a range for the true .

Paired Samples T-Test

Situations for paired samples t-test

Top images from around the web for Situations for paired samples t-test

1 of 2

Top images from around the web for Situations for paired samples t-test

1 of 2
  • Compares two related or dependent samples (before-after measurements, matched pairs)
  • Determines if the mean difference between paired observations significantly differs from zero
  • Frequently used in pre-post study designs (weight before and after a diet program)
  • Compares two different methods on the same subjects (two blood pressure measurement techniques on patients)

Conducting and interpreting paired t-tests

  • Calculate differences between each pair of observations
  • Compute mean difference (dˉ\bar{d}) and standard deviation of differences (sds_d)
  • Calculate t-statistic using formula: t=dˉsd/nt = \frac{\bar{d}}{s_d / \sqrt{n}}
    • nn represents number of paired observations
  • Determine (df) = n−1n - 1
  • Compare calculated t-value to critical t-value at chosen significance level and degrees of freedom
  • If calculated t-value > critical t-value, reject
    • Significant difference exists between paired observations
  • If calculated t-value ≤ critical t-value, fail to reject null hypothesis
    • Insufficient evidence to suggest significant difference between paired observations

Confidence intervals for paired differences

  • Provides range of plausible values for true mean difference
  • Formula: dˉ±tα/2,n−1⋅sdn\bar{d} \pm t_{\alpha/2, n-1} \cdot \frac{s_d}{\sqrt{n}}
    • dˉ\bar{d} represents mean difference
    • tα/2,n−1t_{\alpha/2, n-1} represents critical t-value at chosen confidence level and degrees of freedom
    • sds_d represents standard deviation of differences
    • nn represents number of paired observations
  • Interpretation: (1−α)(1 - \alpha)% confidence that true mean difference falls within calculated interval
  • If excludes zero, suggests significant difference between paired observations (systolic blood pressure before and after medication)

Assumptions of paired samples t-tests

  • Differences between paired observations approximately normally distributed
    • Assess using histogram or normal probability plot
    • Test robust to normality violations for large sample sizes (n > 30)
  • Paired observations independent of each other
    • Each pair should not influence other pairs (multiple measurements on same patient)
  • Data continuous and measured on interval or ratio scale (temperature in ℃, weight in kg)
  • No significant outliers in differences between paired observations
    • Outliers can heavily influence test results (extremely large weight loss in diet study)

Key Terms to Review (18)

Alternative Hypothesis: The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect, a difference, or a relationship in the population. It serves as the focus of research, aiming to provide evidence that supports its claim over the null hypothesis through statistical testing and analysis.
Cohen's d: Cohen's d is a measure of effect size that quantifies the difference between two group means in terms of standard deviation units. It helps in understanding the magnitude of the difference and is especially useful when comparing groups in statistical analyses, such as after performing post-hoc tests or paired samples t-tests. A larger Cohen's d value indicates a more substantial effect or difference, making it easier to interpret the practical significance of statistical findings.
Confidence Interval: A confidence interval is a range of values that is used to estimate an unknown population parameter, calculated from sample data. It provides an interval within which we expect the true parameter to fall with a certain level of confidence, typically expressed as a percentage like 95% or 99%. This concept is fundamental in statistical inference, allowing us to make conclusions about populations based on sample data.
Continuous Data: Continuous data refers to numerical values that can take any value within a given range. This type of data can be measured with precision and often includes measurements such as height, weight, temperature, or time. Continuous data is vital in various statistical analyses and allows for more complex mathematical operations compared to discrete data.
Degrees of Freedom: Degrees of freedom refers to the number of independent values or quantities that can vary in an analysis without breaking any constraints. This concept is crucial in statistical tests because it affects the distribution of the test statistic, influencing how we determine significance. When conducting various statistical tests, understanding degrees of freedom helps in accurately interpreting results and making valid conclusions.
Dependent Samples T-Test: A dependent samples t-test, also known as a paired samples t-test, is a statistical method used to compare the means of two related groups to determine if there is a statistically significant difference between them. This test is ideal for scenarios where the same subjects are measured under two different conditions or at two different times, highlighting the differences in their responses.
Effect Size: Effect size is a quantitative measure that reflects the magnitude of a phenomenon or the strength of the relationship between variables. It helps researchers understand not just whether an effect exists, but how significant that effect is, providing context to statistical results and facilitating comparison across studies. In hypothesis testing, effect size is crucial for interpreting results in relation to practical significance, rather than just statistical significance.
Independence: Independence refers to the condition where two or more events or variables do not influence each other. In statistics, it is a crucial concept that indicates that the occurrence of one event does not affect the probability of another event happening. This idea is foundational in many statistical analyses, including hypothesis testing, regression analysis, and various non-parametric methods.
Matched pairs: Matched pairs refer to a statistical method where two related samples are compared, typically involving subjects that are paired based on specific characteristics to minimize variability. This technique is particularly useful when conducting experiments or analyses where the goal is to determine the effect of a treatment or intervention, as it helps to control for confounding variables by ensuring that each pair shares similar traits. Matched pairs are crucial in improving the accuracy and reliability of results in studies involving comparisons between two groups.
Mean difference: The mean difference is a statistical measure that represents the average of the differences between paired observations in two related groups. It helps in assessing whether there is a significant change or effect between the two sets of data, particularly when analyzing pre-test and post-test results or other paired data comparisons. Understanding the mean difference is essential for interpreting the results of tests that evaluate the effectiveness of interventions or treatments.
Normality: Normality refers to the assumption that the data being analyzed follows a normal distribution, which is a bell-shaped curve characterized by its mean and standard deviation. This concept is crucial as many statistical methods rely on this assumption to provide valid results, impacting hypothesis testing, confidence intervals, and regression analysis.
Null hypothesis: The null hypothesis is a statement that assumes there is no effect or no difference in a given situation, serving as a default position that researchers aim to test against. It acts as a baseline to compare with the alternative hypothesis, which posits that there is an effect or a difference. This concept is foundational in statistical analysis and hypothesis testing, guiding researchers in determining whether observed data can be attributed to chance or if they suggest significant effects.
P-value: A p-value is a statistical measure that helps determine the significance of results from a hypothesis test. It represents the probability of obtaining results at least as extreme as the observed data, given that the null hypothesis is true. A smaller p-value indicates stronger evidence against the null hypothesis, leading to its rejection in favor of an alternative hypothesis.
Paired samples t-test: A paired samples t-test is a statistical method used to determine if there is a significant difference between the means of two related groups. This test is commonly applied in situations where the same subjects are measured before and after an intervention, allowing for direct comparison of changes in outcomes. By accounting for the inherent correlation between paired observations, this test enhances the reliability of conclusions drawn from such data.
R: In statistics, 'r' typically refers to the correlation coefficient, a measure that indicates the strength and direction of a linear relationship between two variables. This value ranges from -1 to 1, where -1 implies a perfect negative correlation, 1 indicates a perfect positive correlation, and 0 suggests no linear relationship. Understanding 'r' is essential when analyzing relationships in various contexts, including decision trees and hypothesis testing.
Sample size: Sample size refers to the number of observations or data points included in a statistical sample, which is crucial for ensuring the reliability and validity of the results. A larger sample size can lead to more accurate estimates and stronger statistical power, while a smaller sample size may result in less reliable outcomes. Understanding the appropriate sample size is essential for various analyses, as it affects the confidence intervals, error rates, and the ability to detect significant differences or relationships within data.
SPSS: SPSS, which stands for Statistical Package for the Social Sciences, is a software application used for statistical analysis and data management. It offers a wide range of statistical tests and procedures, making it an essential tool for researchers and analysts to interpret data efficiently and accurately. Its user-friendly interface allows users to perform complex analyses, such as t-tests, ANOVA, and regression, which connect to various statistical concepts in research methodologies.
T-statistic: A t-statistic is a ratio that compares the difference between the observed sample mean and the hypothesized population mean to the variability of the sample data. It helps determine whether to reject the null hypothesis in hypothesis testing. The t-statistic is particularly useful when sample sizes are small and the population standard deviation is unknown, making it crucial in regression analysis and hypothesis testing.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide