Repeated assessments for a chemical concentration determination of human blood during a
laboratory analysis are known to be normally distributed. Ten assessments on a given sample of
blood yielded the following values:
1.002 0.958 1.014 1.009 1.041 0.962 1.058 1.024 1.019 1.020
Determine a 99% confidence interval for the true chemical concentration in the blood for repeated
assessments of the sample.

To determine the 99% confidence interval for the true chemical concentration in the blood based on the given sample data, we'll follow these steps:

### 1. Compute the sample mean (\(\bar{x}\)):
The sample mean is the average of all the observations.

\[
\bar{x} = \frac{1.002 + 0.958 + 1.014 + 1.009 + 1.041 + 0.962 + 1.058 + 1.024 + 1.019 + 1.020}{10}
\]

### 2. Compute the sample standard deviation (\(s\)):
The sample standard deviation measures the amount of variation or dispersion of the sample data.

\[
s = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}}
\]

Where \(n\) is the number of observations (which is 10 in this case).

### 3. Determine the appropriate t-value:
Since the sample size is small (\(n < 30\)), we'll use the t-distribution rather than the normal distribution. The degrees of freedom (\(df\)) will be \(n - 1 = 9\). For a 99% confidence interval, we'll find the critical t-value (\(t_{\alpha/2}\)) corresponding to \(df = 9\).

### 4. Calculate the margin of error (ME):
The margin of error is calculated as:

\[
ME = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\]

### 5. Determine the confidence interval:
Finally, the 99% confidence interval for the true chemical concentration is given by:

\[
\left(\bar{x} - ME, \bar{x} + ME\right)
\]

Let's calculate the above steps.The 99% confidence interval for the true chemical concentration in the blood based on the given sample data is approximately:

\[
(0.9787, 1.0427)
\]

### Summary of Results:
- **Sample Mean (\(\bar{x}\))**: 1.0107
- **Sample Standard Deviation (\(s\))**: 0.0311
- **t-value**: 3.2498 (for \(df = 9\) and 99% confidence)
- **Margin of Error (ME)**: 0.0320

This interval suggests that we are 99% confident that the true chemical concentration in the blood lies between 0.9787 and 1.0427.

Would you like more details or have any questions?

### Further questions:
1. How does the sample size affect the width of the confidence interval?
2. What would the confidence interval be if we used a 95% confidence level instead?
3. How does the choice of confidence level affect the interval?
4. How would the confidence interval change if the standard deviation were higher?
5. What assumptions are made when using the t-distribution in this context?

### Tip:
Always verify the assumptions behind the statistical methods you use, especially when the sample size is small or the data does not meet the normality assumption.

Learn how to calculate a 99% confidence interval for the true chemical concentration in blood using repeated assessments. Detailed steps include computing the sample mean, sample standard deviation, determining t-values, and calculating the margin of error. Explore the impact of confidence levels and standard deviations on the interval. Ideal for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation