In: Statistics and Probability
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 47 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 = 7.80 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.)
σ is unknown
the distribution of weights is uniform
the distribution of weights is normal
n is large
σ is known
(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 0.01. 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.
(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.)
male firefighters
a)
sample mean, xbar = 37.5
sample standard deviation, σ = 7.8
sample size, n = 47
Given CI level is 99%, hence α = 1 - 0.99 = 0.01
α/2 = 0.01/2 = 0.005, Zc = Z(α/2) = 2.5758
ME = zc * σ/sqrt(n)
ME = 2.5758 * 7.8/sqrt(47)
ME = 2.93
CI = (xbar - Zc * s/sqrt(n) , xbar + Zc * s/sqrt(n))
CI = (37.5 - 2.5758 * 7.8/sqrt(47) , 37.5 + 2.5758 *
7.8/sqrt(47))
CI = (34.57 , 40.43)
lower limit 34.57
upper limit 40.43
margin of error 2.93
b)
the distribution of weights is normal
n is large
σ is known
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 = 2.9, σ = 7.8
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 * 7.8/2.9)^2
n = 48.15
Therefore, the sample size needed to satisfy the condition n
>= 48.15 and it must be an integer number, we conclude that the
minimum required sample size is n = 49
Ans : Sample size, n = 49 or 48
#### If you take z value upto 3 or 4 deciaml answer would be change