Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl). Assume that the population of x values has an approximately normal distribution.
10.10	9.00	10.90	9.10	9.40	9.80	10.00	9.90	11.20	12.10
Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood. (Round your answer to two decimal places.) Find upper and lower limit

To find a 99.9% confidence interval for the population mean of total calcium levels, we will follow these steps:

1. **Calculate the sample mean (\(\bar{x}\))**.
2. **Calculate the sample standard deviation (s)**.
3. **Determine the sample size (n)**.
4. **Find the t-value for 99.9% confidence**.
5. **Calculate the margin of error**.
6. **Construct the confidence interval**.

### Step 1: Calculate the Sample Mean (\(\bar{x}\))
Given the values:
\[ 
10.10, 9.00, 10.90, 9.10, 9.40, 9.80, 10.00, 9.90, 11.20, 12.10 
\]
\[
\bar{x} = \frac{10.10 + 9.00 + 10.90 + 9.10 + 9.40 + 9.80 + 10.00 + 9.90 + 11.20 + 12.10}{10} = \frac{  10.10 + 9.00 + 10.90 + 9.10 + 9.40 + 9.80 + 10.00 + 9.90 + 11.20 + 12.10 }{ 10 }
\]
\[
\bar{x} = \frac{  10.10 + 9.00 + 10.90 + 9.10 + 9.40 + 9.80 + 10.00 + 9.90 + 11.20 + 12.10 }{ 10 } = 10.05 \text{ mg/dl}
\]

### Step 2: Calculate the Sample Standard Deviation (s)
First, calculate the variance:
\[
s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}
\]
Where \( x_i \) are the sample values and \( n \) is the number of samples.

Calculating deviations:
\[
(10.10 - 10.05)^2, (9.00 - 10.05)^2, (10.90 - 10.05)^2, (9.10 - 10.05)^2, (9.40 - 10.05)^2,
\]
\[
(9.80 - 10.05)^2, (10.00 - 10.05)^2, (9.90 - 10.05)^2, (11.20 - 10.05)^2, (12.10 - 10.05)^2
\]

Calculating the squared deviations:
\[
0.0025, 1.1025, 0.7225, 0.9025, 0.4225, 0.0625, 0.0025, 0.0225, 1.3225, 4.2025
\]
Summing them up:
\[
\sum (x_i - \bar{x})^2 = 0.0025 + 1.1025 + 0.7225 + 0.9025 + 0.4225 + 0.0625 + 0.0025 + 0.0225 + 1.3225 + 4.2025 = 8.5
\]
Calculating variance:
\[
s^2 = \frac{8.5}{10-1} = \frac{8.5}{9} \approx 0.9444
\]
Calculating standard deviation:
\[
s \approx \sqrt{0.9444} \approx 0.9724 \text{ mg/dl}
\]

### Step 3: Determine the Sample Size (n)
\[
n = 10
\]

### Step 4: Find the t-value for 99.9% Confidence
Using a t-table for 9 degrees of freedom at a 99.9% confidence level:
\[
t_{0.999, 9} \approx 3.250
\]

### Step 5: Calculate the Margin of Error
\[
ME = t \cdot \left(\frac{s}{\sqrt{n}}\right)
\]
\[
ME = 3.250 \cdot \left(\frac{0.9724}{\sqrt{10}}\right) \approx 3.250 \cdot 0.3077 \approx 1.000
\]

### Step 6: Construct the Confidence Interval
\[
\text{Lower limit} = \bar{x} - ME \approx 10.05 - 1.000 \approx 9.05
\]
\[
\text{Upper limit} = \bar{x} + ME \approx 10.05 + 1.000 \approx 11.05
\]

### Final Result
The 99.9% confidence interval for the population mean of total calcium is approximately:
- **Lower Limit:** 9.05 mg/dl
- **Upper Limit:** 11.05 mg/dl

Would you like to know more details or have any questions? Here are some related questions:
1. How does the t-distribution differ from the normal distribution?
2. What are the implications of a confidence interval in medical studies?
3. How would you interpret a confidence interval of a different confidence level, such as 95%?
4. What factors can affect the width of a confidence interval?
5. How can sample size influence the reliability of a confidence interval?

**Tip:** Always check for normality in your data when using the t-distribution for confidence intervals.

Over the past several months, an adult patient has been treated for tetany (severe muscle spasms). This condition is associated with an average total calcium level below 6 mg/dl. Recently, the patient's total calcium tests gave the following readings (in mg/dl): 10.10, 9.00, 10.90, 9.10, 9.40, 9.80, 10.00, 9.90, 11.20, 12.10. Find a 99.9% confidence interval for the population mean of total calcium in this patient's blood.

Explore how to calculate a 99.9% confidence interval for the population mean of total calcium levels in a patient's blood, based on recent test readings. This guide provides detailed calculations and insights into statistical concepts relevant to medical data analysis.

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