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Uric Acid Overproduction of uric acid in the body can be an indication of cell breakdown. This may be an advance indication of illness such as gout, leukemia, or lymphoma (Reference: Manual of Laboratory and Diagnostic Tests by \(\mathrm{F}\). Fischbach). Over a period of months, an adult male patient has taken eight blood tests for uric acid. The mean concentration was \(\bar{x}=5.35 \mathrm{mg} / \mathrm{d}\). The distribution of uric acid in healthy adult males can be assumed to be normal, with \(\sigma=1.85 \mathrm{mg} / \mathrm{d}\) l. (a) Find a \(95 \%\) confidence interval for the population mean concentration of uric acid in this patient's blood. 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 \(95 \%\) confidence level with maximal margin of error \(E=1.10\) for the mean concentration of uric acid in this patient's blood.

Step by step solution

01

Understanding the Problem

We need to find a 95% confidence interval (CI) for the population mean concentration of uric acid in this patient's blood based on the sample mean, standard deviation, and sample size. Further, we will identify the margin of error and necessary conditions for these calculations. Ultimately, we'll calculate the required sample size for a smaller margin of error.

02

Confidence Interval and Margin of Error (a)

The formula for a confidence interval is \( \bar{x} \pm z^* \frac{\sigma}{\sqrt{n}} \). Given that \( \bar{x} = 5.35 \), \( \sigma = 1.85 \), and \( n = 8 \), first find the critical value \( z^* \) for a 95% confidence level, typically \( 1.96 \). The margin of error (ME) is \( z^* \frac{\sigma}{\sqrt{n}} = 1.96 \times \frac{1.85}{\sqrt{8}} = 1.28 \). Thus, the CI is \( 5.35 \pm 1.28 \), which is (4.07, 6.63).

03

Necessary Conditions (b)

To apply the confidence interval formula, the underlying distribution should be approximately normal or the sample size should be large enough for the Central Limit Theorem to apply. Here, given the normal distribution assumption of uric acid levels, these calculations are valid.

04

Interpretation (c)

This confidence interval indicates that we are 95% confident that the true mean concentration of uric acid in this patient's blood lies between 4.07 mg/dL and 6.63 mg/dL. This range can help assess if the patient's uric acid levels deviate from typical healthy levels.

05

Calculating Sample Size (d)

To achieve a specified margin of error (E) of 1.10 with 95% confidence, use the formula \( n = \left( \frac{z^* \sigma}{E} \right)^2 \). Thus, \( n = \left( \frac{1.96 \times 1.85}{1.10} \right)^2 = 11.34 \). Therefore, at least 12 samples are needed.

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Key Concepts

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

Understanding Margin of Error

The margin of error is a critical component when calculating confidence intervals. It represents the range within which the true population parameter is expected to lie.
The formula used is:

• Margin of Error (ME) = \( z^* \times \frac{\sigma}{\sqrt{n}} \)

Where:

• \( z^* \) is the critical value from the standard normal distribution based on the desired confidence level, here 95%, which is typically 1.96.
• \( \sigma \) is the population standard deviation.
• \( n \) is the sample size.

In our case, the margin of error is calculated as 1.28. This figure gives us an idea of how sensitive our estimate is likely to be to changes in data. By adding and subtracting this margin from the sample mean, we get our confidence interval. For example, with a sample mean of 5.35, our confidence interval spans from 4.07 to 6.63, lending us a precise way to predict where the actual population mean might fall.

Sample Size Calculation

Consider how the sample size influences the reliability of a confidence interval. A larger sample size generally means a smaller margin of error, increasing the precision of our estimates.
To determine the necessary sample size (n) to achieve a specified margin of error, we use the formula:

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

Where:

• \( z^* \) remains the critical value, here 1.96.
• \( \sigma \) is the standard deviation of the population.
• \( E \) is the desired margin of error.

In our challenge, to ensure the margin of error is 1.10, the calculation results in a need for at least 12 samples. Keeping a larger dataset not only enhances accuracy but often is necessary for valid statistical results when data is less than perfectly random.

Normal Distribution Significance

Understanding the role of the normal distribution is paramount for confidence intervals. The normal distribution, often visualized as a bell curve, describes how data points are dispersed around the mean.
When data is normally distributed, we can use standard normal distribution tables to decide the critical values, like \( z^* \), crucial for margin of error and confidence interval calculations.
Why is this important?

• A standard normal distribution enables predictions about where data is expected to fall.
• It facilitates robust statistical testing and inferential statistics.
• For uric acid levels, the assumption of normal distribution simplifies calculations and yields straightforward interpretations.

Even if data is not perfectly normal, as long as the sample size is sufficiently large, the Central Limit Theorem tells us that the distribution of the sample mean will approximate a normal distribution.