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 48 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 = 8.00 ml/kg for the distribution of blood plasma.
Find the sample size necessary for a 99% confidence level with maximal error of estimate E = 2.40 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)
male firefighters
To find the required sample size for a 99% confidence level with maximum error of estimate E = 2.40 for the mean plasma volume in male firefighters, we can use the formula for sample size determination:
n = (Z * σ / E)²
Where:
n = required sample size
Z = Z-score corresponding to the desired confidence level
σ = standard deviation of the population (blood plasma in this case)
E = maximum error of estimate (also known as margin of error)
In this case, Z-score for a 99% confidence level is 2.576 (you can find this value from a standard normal distribution table or using a calculator). σ (standard deviation) is given as 8.00 ml/kg and E (maximum error of estimate) is 2.40 ml/kg.
Substituting the given values into the formula:
n = (2.576 * 8.00 / 2.40)²
Calculating this expression:
n = (20.608 / 2.40)²
n = 8.58778²
n ≈ 73.665
Rounding up to the nearest whole number, the required sample size is approximately 74 male firefighters.