Question

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 44 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 = 7.10 ml/kg for the distribution of blood plasma.

(a) Find a 99% confidence interval for the population mean blood plasma volume in male firefighters. What is the margin of error? (Round your answers to two decimal places.)

lower limit
upper limit
margin of error

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

the distribution of weights is uniformσ is unknownσ is knownn is largethe distribution of weights is normal

(c) Interpret your results in the context of this problem.

We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.

We are 1% confident that the true average blood plasma volume in male firefighters falls within this interval.   

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.99.

The probability that this interval contains the true average blood plasma volume in male firefighters is 0.01.

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.90 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)

Answer 1

a)
sample mean, xbar = 37.5
sample standard deviation, σ = 7.1
sample size, n = 44

Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.58

ME = zc * σ/sqrt(n)
ME = 2.58 * 7.1/sqrt(44)
ME = 2.76

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (37.5 - 2.58 * 7.1/sqrt(44) , 37.5 + 2.58 * 7.1/sqrt(44))
CI = (34.74 , 40.26)

lower limit = 34.74 ,
upper limit = 40.26
margin of error = 2.76

b)
σ is known
the distribution of weights is normal

c)
We are 99% confident that the true average blood plasma volume in male firefighters falls within this interval.

d)
The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 2.9, σ = 7.1

The critical value for significance level, α = 0.01 is 2.58.

The following formula is used to compute the minimum sample size required to estimate the population mean μ within the required margin of error:
n >= (zc *σ/E)^2
n = (2.58 * 7.1/2.9)^2
n = 39.9

n = 40