Total plasma volume is important in determining the required plasma component in blood replacement theory for a person undergoing surgery. Plasma volume is influenced by the overall health and physical activity of an individual. Suppose that sample of some male firefighters are tested and that they have a plasma volume sample mean of = 36.5 ml/kg (milliliters of plasma per kilogram body weight). Assume that = 7.60 ml/kg for the distribution of blood plasma. Find the smallest sample size necessary for a 98% confidence level with maximal error estimate E = 3.1.

Answer
a. 29
b. 33
c. 37
d. 3
e. 6

To find the smallest sample size necessary for a 98% confidence level with a maximal error estimate of E = 3.1, we can use the formula for sample size calculation in estimating a population mean.

The formula is given by:

n = (Z * σ / E)^2

Where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (98% in this case)
- σ is the standard deviation of the population
- E is the maximal error estimate

In this case, we are given that the sample mean is 36.5 ml/kg (sample mean = x̄) and the distribution of blood plasma (population standard deviation = σ) is 7.60 ml/kg.

First, we need to find the Z-score corresponding to the 98% confidence level. The Z-score can be found using a Z-table or a statistical calculator. For a 98% confidence level, the Z-score is approximately 2.33.

Now, we can substitute the given values into the formula:

n = (Z * σ / E)^2
n = (2.33 * 7.60 / 3.1)^2
n = (17.608 / 3.1)^2
n = 5.68^2
n ≈ 32.2624

Since we need to find the smallest sample size necessary, we round up the calculated value to the nearest whole number:

n ≈ 33

Therefore, the smallest sample size necessary for a 98% confidence level with a maximal error estimate of E = 3.1 is 33. Hence, the answer is b. 33.