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 41 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 = 7.60 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 normal

σ is known

σ is unknown

the distribution of weights is uniform

n is large

(c) Interpret your results in the context of this problem. (Select all that apply.)

99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.

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

1% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.

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 = 2.60 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)
___________ male firefighters

Answer 1

Part a)

Confidence Interval :-
X̅ ± Z( α /2) σ / √ ( n )
Z(α/2) = Z (0.01 /2) = 2.576
37.5 ± Z (0.01/2 ) * 7.6/√(41)
Lower Limit = 37.5 - Z(0.01/2) 7.6/√(41)
Lower Limit = 34.4425 ≈ 34.44
Upper Limit = 37.5 + Z(0.01/2) 7.6/√(41)
Upper Limit = 40.5575 ≈ 40.56
99% Confidence interval is ( 34.44 , 40.56 )

Margin of Error = Z (0.01/2 ) * 7.6/√(41) = 3.0575 ≈ 3.06

Part b)

The distribution of weights is normal

σ is known

n is large

Part c)

99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters.

Part d)

Sample size can be calculated by below formula
n = (( Z(α/2) * σ) / e )²
n = (( Z(0.01/2) * 7.6 ) / 2.6 )²
Critical value Z(α/2) = Z(0.01/2) = 2.5758
n = (( 2.5758 * 7.6 ) / 2.6 )²
n = 57
Required sample size at 99% confident is 57.