Estimating Sample Size For Population Mean With Known Standa

Estimating Sample Size for Population Mean with Known Standard Deviation

Estimating Sample Size for Population Mean with Known Standard Deviation

In statistical analysis, determining the appropriate sample size is crucial for achieving reliable and accurate estimates of population parameters. When estimating the average amount a family spends on food annually, especially with known population standard deviation, planning involves using the formula derived from the properties of the normal distribution and confidence intervals.

Suppose the historical data indicates a population standard deviation of $1200 in family food expenditures. To estimate the true population mean within a margin of error of $60 with a confidence level of 99%, a researcher must compute the minimum sample size required.

The formula for calculating the necessary sample size (n) in this scenario is:

n = (Zα/2 * σ / E)^2

where:

  • Zα/2 is the z-score corresponding to the desired confidence level (for 99%, Zα/2 ≈ 2.576)
  • σ is the known population standard deviation ($1200)
  • E is the margin of error ($60)

Plugging in the values:

n = (2.576 * 1200 / 60)^2

Calculating step by step:

  1. Compute numerator: 2.576 * 1200 = 3091.2
  2. Divide by E: 3091.2 / 60 = 51.519999...
  3. Square the result: 51.52^2 ≈ 2662.78

Since the sample size must be a whole number, we round up to the nearest whole number:

n ≈ 2663

Therefore, at least 2,663 families need to be surveyed to ensure the estimated average food expenditure is within $60 of the true mean with 99% confidence.

References

  • Creswell, J. W. (2021). Research design: Qualitative, quantitative, and mixed methods approaches. Sage publications.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2021). Introduction to the Practice of Statistics. W.H. Freeman and Company.
  • Devore, J. L. (2018). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2019). Statistics for Business and Economics. Pearson.
  • LaMorte, W. (2020). Sample size calculations. Boston University School of Public Health. https://sphweb.bumc.bu.edu/
  • Levine, D. M., Stephan, D. F., Krehbiel, T. C., & Berenson, M. L. (2018). Statistics for Managers Using Microsoft Excel. Pearson.
  • Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
  • Ott, R. L., & Longnecker, M. (2021). An Introduction to Statistical Methods and Data Analysis. Cengage Learning.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Ghasemi, A., & Zahediasl, S. (2012). Normality tests for statistical analysis: a guide for non-statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486-489.

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