Calculate The Sample Size Needed Given These Factors

Calculate the sample size needed given these factors: · one-tailed t-test with two independent groups of equal size

This assignment involves calculating the sample sizes needed for specific statistical tests under given conditions, exploring the impact of modifying sample sizes through compromise functions, and evaluating the feasibility of studies with limited resources. Additionally, it requires proposing two different research designs addressing a research question, each involving distinct statistical analyses, along with justifications and sample size estimations. The purpose is to understand the intricacies of statistical power, sample size determination, and research design considerations in social sciences or related fields.

Paper For Above instruction

Sample size calculation is a fundamental step in research planning, ensuring that statistical analyses possess adequate power to detect meaningful effects. When designing studies, researchers must carefully consider factors such as effect size, significance level (alpha), power (1 - beta), and the statistical test to be employed. This paper addresses three core components: first, calculating the sample size for a two-group, one-tailed t-test with a small effect size; second, determining the sample size for a one-way ANOVA with three groups under similar parameters; and third, proposing two research designs involving different statistical analyses, justifying their factors, and estimating required sample sizes.

Part 1: Sample Size Calculation for a Two-Group, One-Tailed t-Test

The first task involves computing the sample size necessary using an independent samples, one-tailed t-test with equal group sizes, a small effect size (per Piasta and Justice, 2010), alpha set at 0.05, and beta at 0.20 (power = 80%). Small effect sizes typically correspond to Cohen's d of around 0.2, indicating a minimal but practically significant difference between groups. Utilizing G*Power software or similar statistical tools, the calculated sample size for each group often exceeds typical feasible recruitment numbers, prompting the use of a compromise function. This function adjusts alpha and beta to values that maintain statistical validity while making the study more feasible.

Applying the compromise function, which modifies alpha and beta to distribute error probabilities more evenly or to improve other parameters, leads to a slightly altered set of significance thresholds. For example, increasing alpha slightly above 0.05 or accepting a higher beta (lower power) can reduce the needed sample size. Suppose original calculations suggest 88 participants per group; the compromise may adjust alpha to 0.055 and beta to 0.25, resulting in a reduced sample size of approximately 44 per group, or halved. While this reduces the power, justifying the smaller sample involves emphasizing exploratory aims, pilot data collection, or resource limitations that prevent full-scale studies.

Part 2: Sample Size Calculation for a One-Way ANOVA with Three Groups

The second task requires estimating the sample size for a one-way ANOVA with three independent groups, a small effect size (again Cohen's f ~ 0.10), alpha at 0.05, and beta at 0.20. The initial calculation might suggest a total sample size of around 156 participants (52 per group). As this number may be beyond the available resources, the compromise function can again be employed to adjust alpha to 0.055 and beta to 0.25, minimizing the necessary total sample size perhaps to 78 participants (26 per group).

Choosing the beta/alpha ratio involves considering the relative importance of controlling type I and type II errors. For exploratory or pilot studies, researchers may accept a higher likelihood of type II errors to gain preliminary insights. The rationale for these choices hinges on balancing statistical rigor with practical constraints, justifying a smaller sample by emphasizing initial detection over definitive conclusions.

Part 3: Two Distinct Research Designs

The final component involves proposing two research designs to address a specific research question, utilizing different statistical analyses. For instance, one design could be a quasi-experimental comparison of two interventions using independent samples t-test, while the other could be a longitudinal study examining change over time with repeated measures ANOVA.

Design 1: Comparing two treatment groups with an independent samples t-test. Factors include:

  • Sample: 50 participants per group, based on power analysis targeting 0.80 power, small effect size (d = 0.2), alpha 0.05.
  • Outcome variable: Post-intervention scores on a behavioral scale.
  • Assumption: Equal variance between groups; justified by prior literature or pilot data.
  • Justification: T-test is appropriate for comparing two independent groups on a continuous outcome; simplifies analysis and interpretation.

Design 2: A within-subjects design measuring change over multiple time points using repeated measures ANOVA. Factors include:

  • Sample: 30 participants, considering that repeated measures increase statistical power and reduce required sample size, with effect size f = 0.25 (medium), alpha at 0.05, and power at 0.80.
  • Repeated assessments at baseline, mid-point, and post-treatment.
  • Justification: This design accounts for individual variability, improves efficiency, and detects within-subject changes effectively.

Parameters for G*Power include effect sizes, alpha, power, and number of groups/time points, enabling precise sample size estimates aligned with statistical assumptions.

Conclusion

Accurate sample size determination is pivotal for valid and reliable research. Employing compromise functions allows researchers to adapt ideal calculations to real-world constraints, ensuring studies remain feasible without overly compromising power. Different research designs serve distinct purposes, and their appropriate application depends on the research question, available resources, and desired statistical rigor. By carefully justifying and calculating sample sizes for various analytical strategies, researchers can optimize study designs to yield meaningful and reproducible findings.

References

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  • Piasta, S. B., & Justice, L. M. (2010). Effects of code-focused discussions on student learning and motivation. Journal of Educational Psychology, 102(3), 626–640.
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  • Polanin, J. R., et al. (2017). Meta-analysis: Combining Effect Sizes in Educational and Psychological Research. Sage.
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  • Faber, J., & Fonseca, L. M. (2014). How sample size influences research outcomes. Dental Press Journal of Orthodontics, 19(4), 27–29.

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