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 45 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 = 7.30 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.)

1. lower limit =

2. upper limit =

3. margin of error =   

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

- the distribution of weights is uniform

- σ is known

- the distribution of weights is normal

- σ is unknown

- n is large

(c) Interpret your results in the context of this problem. Choose 1:

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

- 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.

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

(d) Find the sample size necessary for a 99% confidence level with maximal margin of error E = 2.80 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.)

1. Male firefighters =

Answer 1

a)

sample mean, xbar = 37.5
sample standard deviation, σ = 7.3
sample size, n = 45

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 * 7.3/sqrt(45)
ME = 2.81

CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (37.5 - 2.58 * 7.3/sqrt(45) , 37.5 + 2.58 * 7.3/sqrt(45))
CI = (34.69 , 40.31)

1. lower limit = 34.69

2. upper limit = 40.31

3. margin of error = 2.81

b)

σ is known

- 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 = 2.8, σ = 7.3

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 * 7.3/2.8)^2
n = 45.24

Therefore, the sample size needed to satisfy the condition n >= 45.24 and it must be an integer number, we conclude that the minimum required sample size is n = 46
Ans : Sample size, n = 46