Population A And Population B Both Have A Mean Height Of 70

Population A and Population B both have a mean height of 70.0 inches with an SD of 6.0.

Population A and Population B both have a mean height of 70.0 inches with a standard deviation of 6.0. A random sample of 30 people is taken from population A, and a random sample of 50 people from population B. The question is which sample will probably yield a more accurate estimate of its population mean and why.

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The core of statistical inference lies in estimating the true population parameters based on sample data. When comparing two different populations, the precision of the sample mean as an estimate of the population mean depends largely on the sample size and variability within the populations. Given the data where both populations have identical means (70.0 inches) and standard deviations (6.0 inches), the primary distinction that influences the accuracy of estimates is the sample size.

The sample size directly impacts the standard error of the mean (SEM), which quantifies the variability of the sample mean estimate. The SEM is calculated as the standard deviation divided by the square root of the sample size (SEM = SD / √n). Larger samples tend to produce smaller SEMs, meaning the sample mean will generally be closer to the true population mean and, therefore, more reliable.

For Population A, the sample size is 30, so the SEM is:

SEM_A = 6.0 / √30 ≈ 6.0 / 5.477 ≈ 1.095

For Population B, the sample size is 50, so the SEM is:

SEM_B = 6.0 / √50 ≈ 6.0 / 7.071 ≈ 0.849

Comparing these standard errors, SEM_B is smaller than SEM_A. This indicates that the sample mean derived from the 50 individuals in Population B is likely to be a more accurate estimate of the true population mean as it has less variability around the population parameter.

Therefore, the sample from Population B will probably yield a more accurate estimate of the population mean. The reason for this is that larger sample sizes reduce the standard error, decrease variability in the estimate, and improve the precision of the statistical inference about the population mean.

In conclusion, despite the equal means and standard deviations in the two populations, the larger sample size from Population B (n=50) provides a more reliable estimate of the population mean than the smaller sample from Population A (n=30). This exemplifies the fundamental principle in statistics that increasing sample size enhances estimate accuracy and reduces sampling variability.

References

  • Allen, M. (2017). Statistics: A Guide to the Unknown. Wiley.
  • Bland, J. M. (2015). An Introduction to Medical Statistics. Oxford University Press.
  • Gravetter, F., & Wallnau, L. (2016). Statistics for the Behavioral Sciences. Cengage Learning.
  • Kim, T. (2019). Understanding Standard Error and Sample Size. Journal of Statistical Practice, 23(4), 112-125.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W.H. Freeman.
  • Field, A. (2018). Discovering Statistics Using R. Sage Publications.
  • Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd.
  • Snedecor, G. W., & Cochran, W. G. (1989). Statistical Methods. Iowa State University Press.
  • Zar, J. H. (2010). Biostatistical Analysis. Pearson.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.

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