Total plasma volume is important in determining the required plasma component in
ID: 3154995 • Letter: T
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 50 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 = 8.00 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.) is unknown n is large is known the distribution of weights is uniform the distribution of weights is normal Correct: Your answer is correct. (c) Interpret your results in the context of this problem. 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 0.99. 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 = 2.30 for the mean plasma volume in male firefighters. (Round up to the nearest whole number.) male firefighters
Explanation / Answer
a)
Note that
Margin of Error E = z(alpha/2) * s / sqrt(n)
Lower Bound = X - z(alpha/2) * s / sqrt(n)
Upper Bound = X + z(alpha/2) * s / sqrt(n)
where
alpha/2 = (1 - confidence level)/2 = 0.005
X = sample mean = 37.5
z(alpha/2) = critical z for the confidence interval = 2.575829304
s = sample standard deviation = 8
n = sample size = 50
Thus,
Margin of Error E = 2.914218188
Lower bound = 34.58578181
Upper bound = 40.41421819
Thus, the confidence interval is
( 34.58578181 , 40.41421819 ) [ANSWER]
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b)
We assumed that the sample size is large enough for the sampling distribution of the mean to be approximately normally distributed.
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c)
OPTION A: 99% of the intervals created using this method will contain the true average blood plasma volume in male firefighters. [ANSWER]
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d)
Note that
n = z(alpha/2)^2 s^2 / E^2
where
alpha/2 = (1 - confidence level)/2 = 0.005
Using a table/technology,
z(alpha/2) = 2.575829304
Also,
s = sample standard deviation = 8
E = margin of error = 2.3
Thus,
n = 80.27096077
Rounding up,
n = 81 [ANSWER]
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