Questions: 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 40 male firefighters are tested and that they have a plasma volume sample mean of x̄=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 34.41 upper limit 40.59 margin of error 3.09 (b) What conditions are necessary for your calculations? (Select all that apply.) σ is unknown n is large the distribution of volumes is normal the distribution of volumes is uniform σ is known

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 40 male firefighters are tested and that they have a plasma volume sample mean of x̄=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
34.41
upper limit
40.59
margin of error
3.09
(b) What conditions are necessary for your calculations? (Select all that apply.)
σ is unknown
n is large
the distribution of volumes is normal
the distribution of volumes is uniform
σ is known

Transcript text: 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 40 male firefighters are tested and that they have a plasma volume sample mean of $\bar{x}=37.5 \mathrm{ml} / \mathrm{kg}$ (milliliters plasma per kilogram body weight). Assume that $\sigma=7.60 \mathrm{ml} / \mathrm{kg}$ 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 34.41 upper limit 40.59 margin of error 3.09 (b) What conditions are necessary for your calculations? (Select all that apply.) $\sigma$ is unknown $n$ is large the distribution of volumes is normal the distribution of volumes is uniform $\sigma$ is known

Solution

Solution Steps

Step 1: Calculate the Confidence Interval

To find the $99\%$ confidence interval for the population mean blood plasma volume, we use the formula:

\[ \bar{x} \pm z \frac{\sigma}{\sqrt{n}} \]

Where:

$\bar{x} = 37.5 \, \text{ml/kg}$ (sample mean)
$z = 2.58$ (Z-score for $99\%$ confidence level)
$\sigma = 7.6 \, \text{ml/kg}$ (population standard deviation)
$n = 40$ (sample size)

Calculating the margin of error:

\[ \text{Margin of Error} = z \cdot \frac{\sigma}{\sqrt{n}} = 2.58 \cdot \frac{7.6}{\sqrt{40}} \approx 3.1 \]

Thus, the confidence interval is:

\[ 37.5 \pm 3.1 = (34.4, 40.6) \]

Step 2: State the Results

The $99\%$ confidence interval for the population mean blood plasma volume in male firefighters is:

Lower Limit: $34.4 \, \text{ml/kg}$
Upper Limit: $40.6 \, \text{ml/kg}$
Margin of Error: $3.1 \, \text{ml/kg}$

Step 3: Identify Necessary Conditions

The conditions necessary for the calculations are:

$n$ is large
$\sigma$ is known

Final Answer

The $99\%$ confidence interval is $(34.4, 40.6)$ with a margin of error of $3.1$. The necessary conditions are that $n$ is large and $\sigma$ is known.

\[ \boxed{\text{Lower Limit: } 34.4, \text{ Upper Limit: } 40.6, \text{ Margin of Error: } 3.1} \]

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