Question unclear. Without further information about the design and planned test, it is impossible to say anything. You could have a simple t-test or a multilevel structural equation model.

I am not aware of that formula. However, I have used the Krejcie and Morgan formula (1970) and the appropriate sample size for your universe is 380.

https://nursesrevisionuganda.com/sample-size-determination/

General Theory Behind Sample Sizes and Populations The policy of our Institute is "Advancing Research through Teaching and Learning". cut and paste, or solving specific problems, should only be for you to solve, or do your homework, as evidence of your learning. Therefore, I assume that your question is either a request to compare your results to see how close your answer was, or it is a request to understand the relevancy of sample size as a fraction of the target population or census. That said, let's discuss the basics of sampling to make this worthwhile for both of us.

Sample Size Representation:

In research, the sample size must adequately represent the population to ensure that the findings are generalizable. The best sample size is the census, but not everybody has access to a supercomputer, and we cannot scan people (yet). Therefore, samples are our best short A sample that is too small may not capture the variability within the population, while a sample that is too large may be unnecessary and inefficient. The size of the sample is often determined based on the population size, the desired confidence level, and the margin of error, hence the sampling methodology is a critical factor in ensuring the sample is a good fit the purpose of the study. Pretest and Phenomenon Inclusion: In some cases, a purposive sample or a pretest can be used to ensure that specific phenomena of interest are included in the sample. This is especially relevant when studying rare events or subgroups within a population. By ensuring these subgroups are represented, researchers can increase the validity and relevance of their findings. Degrees of Freedom and Probability: Degrees of freedom (df) refer to the number of independent values that can vary in an analysis without violating any constraints. As sample sizes increase, degrees of freedom also increase, leading to more precise estimates of population parameters. This precision, in turn, improves the probability of detecting true effects (higher power) and reduces the likelihood of Type I (false positive) or Type II (false negative) errors. This relationship becomes especially significant in hypothesis testing, where larger samples offer higher probabilities of correctly rejecting null hypotheses. Sample Size and Correlation Analysis: In correlation analysis, where the goal is to assess the relationship between two or more variables, larger samples provide more reliable estimates of correlation coefficients. The larger the sample size, the closer the sample correlation will be to the true population correlation. A small sample may result in spurious correlations, while a sufficiently large sample provides a clearer picture of the true relationship. Sample Size in Exploratory vs. Confirmatory Studies:In exploratory studies, where the aim is to discover patterns and relationships, a smaller, purposive sample may be acceptable. However, caution must be exercised because such samples are less generalizable. In confirmatory studies, larger sample sizes are critical to ensure the statistical power necessary to test hypotheses with confidence. Degrees of Freedom and Causal Determination: Degrees of freedom play a critical role in regression and other models where causal relationships are inferred. As the sample size increases, the model’s ability to accurately estimate parameters improves, increasing the probability of correctly identifying causal relationships between variables.Application of Taro Yamane's Formula Taro Yamane's formula is commonly used to determine sample size based on population size. The formula is given as:

n=N/1+N(e2)

n = N/1 + N(e^2)

n=N/1+N(e2)

n = sample size N = population size e = margin of error (usually set at 0.05 for a 95% confidence level) Given: Population size (N) = 39,789 students Margin of error (e) = 0.05 Calculation of sample size using Taro Yamane’s formula follows.

n=39,789/1+39,789(0.052)

n=1+39,789(0.052)39,789​

n=39,789/1+39,789(0.0025)

n = 39,789/1 + 39,789(0.0025)

n=39,789/1+99.4725

n=39,789/1+99.472539,789​

n=39,789/100.4725≈396.1n = 39,789/100.4725 = approximately 396.1n=Thus, the required size for generalization is approximately 396 students.

Final WORD: In quantitative research, determining an appropriate sample size is crucial for ensuring reliable and valid results. Parameters like degrees of freedom, correlation analysis, and causal determination, researchers can make informed decisions about sample size by aligning sample size requirements to specifics of your inquiry is the true determinant of a statistically significant sample size. Good Luck.

References:

Israel, G. D. (1992). Determining sample size. University of Florida Cooperative Extension Service, Institute of Food and Agriculture Sciences, EDIS. https://www.tarleton.edu/academicassessment/documents/Samplesize.pdf

Yamane, T. (1967). Statistics: An introductory analysis (2nd ed.). New York: Harper and Row.