Total Plasma Volume Is Important In Determining The Required

Total Plasma Volume Is Important In Determining The Required Plasma Co

Total plasma volume is important in determining the required plasma component in blood replacement therapy for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that a random sample of 46 male firefighters are tested and that they have a plasma volume sample mean of x = 37.5 ml/kg (milliliters plasma per kilogram body weight). Assume that σ = 7.00 ml/kg for the distribution of blood plasma.

(a) Find an 80% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Use 2 decimal places.)

Given data: sample size n = 46, sample mean x̄ = 37.5 ml/kg, population standard deviation σ = 7.00 ml/kg, confidence level = 80%

To compute the confidence interval, we use the z-distribution since σ is known.

First, find the z-score corresponding to an 80% confidence level. The z-value for an 80% confidence interval (which leaves 10% in each tail) is approximately 1.28 (using standard normal distribution tables or calculator).

Calculate the standard error: SE = σ / √n = 7.00 / √46 ≈ 7.00 / 6.782 ≈ 1.033 ml/kg

Calculate the margin of error (MOE): MOE = z SE = 1.28 1.033 ≈ 1.32 ml/kg

Determine the confidence interval limits:

  • Lower limit = x̄ - MOE = 37.5 - 1.32 ≈ 36.18 ml/kg
  • Upper limit = x̄ + MOE = 37.5 + 1.32 ≈ 38.82 ml/kg

Thus, the 80% confidence interval for the mean plasma volume is approximately (36.18 ml/kg, 38.82 ml/kg). The margin of error is approximately 1.32 ml/kg.

(b) What conditions are necessary for your calculations? (Select all that apply.)

  • The distribution of weights is normal
  • σ is known
  • n is large

Explanation: For the z-interval to be valid, the population distribution should be normal or the sample size large enough (n ≥ 30) for the Central Limit Theorem to apply. Given that σ is known, these are the key conditions.

(c) Give a brief interpretation of your results in the context of this problem.

The 80% confidence interval indicates that we can be 80% confident that the true average blood plasma volume for all male firefighters lies between approximately 36.18 ml/kg and 38.82 ml/kg. This means that, based on our sample, the average plasma volume in this population is likely within this interval, which can help in planning blood replacement therapies during surgeries.

(d) Find the sample size necessary for an 80% confidence level with a maximal error of estimate E = 2.00 ml/kg.

To determine the required sample size, we use the formula:

n = (z * σ / E)^2

We already know: z = 1.28 (for 80% confidence), σ = 7.00 ml/kg, E = 2.00 ml/kg.

Calculations:

n = (1.28 * 7.00 / 2.00)^2 = (8.96 / 2.00)^2 = (4.48)^2 ≈ 20.07

Since sample size must be a whole number, we round up to the next whole number: n ≈ 21.

Therefore, a sample size of at least 21 firefighters would be necessary to achieve the desired margin of error with 80% confidence.

References

  • Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences (4th ed.). Pearson.
  • Bluman, A. G. (2018). Elementary Statistics: A Step By Step Approach ( nineh edition). McGraw-Hill Education.
  • Dean, A., & Voss, D. (2010). Design and Analysis of Experiments. Springer.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W.H. Freeman.
  • Newcombe, R. G. (1998). Two-sided confidence intervals for the single proportion: Comparison of seven methods. Statistics in Medicine, 17(8), 857-872.
  • Portney, L. G., & Watkins, M. P. (2015). Foundations of Clinical Research: Applications to Practice (3rd ed.). F.A. Davis Company.
  • Schmidt, R., & Hunter, J. (2014). Methods of Meta-Analysis: Correcting Error and Bias in Research Findings. Sage Publications.
  • Schleifer, M. (2015). Basic Concepts of Sampling. In Sampling: Design and Analysis (pp. 45-58). Springer.
  • Venkataraman, S., & Tannock, I. (2017). Confidence Intervals and Their Application in Clinical Research. Clinical Oncology, 29(4), 241-246.
  • Zar, J. H. (2010). Biostatistical Analysis (5th ed.). Pearson.

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