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 = 7.10 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 uniform

σ is unknown

σ is known

n is large

the distribution of weights is normal

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

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 0.99.  

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 0.01.

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.40 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)
male firefighters

Answer 1

a)

sample mean, xbar = 37.5
sample standard deviation, σ = 7.1
sample size, n = 41

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(41)
ME = 2.86

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (37.5 - 2.58 * 7.1/sqrt(41) , 37.5 + 2.58 * 7.1/sqrt(41))
CI = (34.64 , 40.36)

Lower limit = 34.64
Upper limit = 40.36
Margin of error = 2.86

b)

σ is known

n is large

the distribution of weights is normal

c)

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

d)

The following information is provided,
Significance Level, α = 0.01, Margin or Error, E = 2.4, 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 * 7.1/2.4)^2
n = 58.26

Therefore, the sample size needed to satisfy the condition n >= 58.26 and it must be an integer number, we conclude that the minimum required sample size is n = 59
Ans : Sample size, n = 59 or 58

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If you take z value upto 3 or 4 decimal answer would be change