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Plasma Volume 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. (Reference: See Problem 16.) Suppose that a random sample of 45 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 \(\sigma=7.50 \mathrm{ml} / \mathrm{kg}\) 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? (b) What conditions are necessary for your calculations? (c) Interpret Compare your results in the context of this problem. (d) Sample Size Find the sample size necessary for a \(99 \%\) confidence level with maximal margin of error \(E=2.50\) for the mean plasma volume in male firefighters.

Step by step solution

01

Identify the given information

We are given the sample size \(n=45\), sample mean \(\bar{x}=37.5 \, \text{ml/kg}\), population standard deviation \(\sigma=7.5 \, \text{ml/kg}\), and we need a \(99\%\) confidence interval for the population mean.

02

Find the z-score for 99% confidence

For a \(99\%\) confidence interval, the z-score corresponding to the tail probability of \(0.5\%\) is approximately \(z=2.576\).

03

Calculate the standard error

The standard error (SE) of the sample mean is given by \(SE = \frac{\sigma}{\sqrt{n}} = \frac{7.5}{\sqrt{45}} \approx 1.118\).

04

Calculate the margin of error

The margin of error (ME) is calculated using \(ME = z \times SE = 2.576 \times 1.118 \approx 2.88\).

05

Determine the confidence interval

The confidence interval is calculated using \(\bar{x} \pm ME = 37.5 \pm 2.88\), resulting in \([34.62, 40.38]\, \text{ml/kg}\).

06

State the conditions for calculation

The conditions required are: the sample is randomly selected, the population distribution is approximately normal, and the sample size is sufficiently large for the Central Limit Theorem to apply.

07

Interpret the results

We are \(99\%\) confident that the true mean plasma volume for male firefighters is between \(34.62\, \text{ml/kg}\) and \(40.38\, \text{ml/kg}\).

08

Calculate sample size for given error

Use the formula \(n = \left(\frac{z \cdot \sigma}{E}\right)^2\). With \(z=2.576\), \(\sigma=7.5\), and \(E=2.5\), we get \(n = \left(\frac{2.576 \times 7.5}{2.5}\right)^2 \approx 59.9\). Hence, \(n \approx 60\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Size Determination

When planning a study or experiment, determining the appropriate sample size is crucial. This step ensures that the results will have the desired level of precision and reliability. The goal here is to ascertain a sample size that will provide enough power to detect a true effect or mean, while keeping the margin of error at a specified level for a given confidence level.

To calculate the sample size needed for a specific margin of error, use the formula:

• \[ n = \left(\frac{z \cdot \sigma}{E}\right)^2 \]

where:

• \(n\) is the sample size,
• \(z\) is the z-score corresponding to the desired confidence level,
• \(\sigma\) is the population standard deviation,
• \(E\) is the maximum allowable margin of error.

In our example, with a 99% confidence level and a margin of error of 2.5 ml/kg, the calculated sample size was approximately 60. This means, to ensure accuracy and reliability, the study should include around 60 male firefighters.

Population Mean

Understanding the population mean is key whenever conducting statistical analysis. The population mean is the average value of a particular characteristic within the entire population. It represents the central point of the data set.

However, in many cases, calculating the population mean directly is impractical or impossible due to the size and scope of the population. Instead, a sample mean, denoted as \(\bar{x}\), is used as an estimate. In the context of our problem, the sample mean plasma volume was calculated at 37.5 ml/kg based on a sample of male firefighters.

A sample provides a practical way to estimate the population mean using collected data while also allowing for the application of inferential statistics, such as confidence intervals.

Margin of Error

The margin of error is a critical part of statistical analysis as it indicates the amount of uncertainty in the estimate of the population parameter. It tells us how far the sample mean could deviate from the true population mean. This concept is often expressed in terms of a range, which allows us to express the confidence level associated with the data.

To calculate the margin of error, the formula used is:

• \[ ME = z \times \text{SE} \]

where:

• \(ME\) is the margin of error,
• \(z\) is the z-score for the confidence level,
• \(\text{SE}\) is the standard error defined as \(\frac{\sigma}{\sqrt{n}}\).

In our example, for a 99% confidence interval, the calculated margin of error was approximately 2.88 ml/kg. This means, we can say with 99% confidence, the true mean plasma volume for the population is within 2.88 ml/kg of the sample mean, both above and below.