Math Problem Statement
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
Solution
To find a 99.9% confidence interval for the population mean of total calcium levels, we will follow these steps:
- Calculate the sample mean ().
- Calculate the sample standard deviation (s).
- Determine the sample size (n).
- Find the t-value for 99.9% confidence.
- Calculate the margin of error.
- Construct the confidence interval.
Step 1: Calculate the Sample Mean ()
Given the values:
Step 2: Calculate the Sample Standard Deviation (s)
First, calculate the variance: Where are the sample values and is the number of samples.
Calculating deviations:
Calculating the squared deviations: Summing them up: Calculating variance: Calculating standard deviation:
Step 3: Determine the Sample Size (n)
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:
Step 5: Calculate the Margin of Error
Step 6: Construct the Confidence Interval
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:
- How does the t-distribution differ from the normal distribution?
- What are the implications of a confidence interval in medical studies?
- How would you interpret a confidence interval of a different confidence level, such as 95%?
- What factors can affect the width of a confidence interval?
- 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.
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Math Problem Analysis
Mathematical Concepts
Statistics
Confidence Intervals
Normal Distribution
Formulas
Sample Mean (\bar{x}) = \frac{\sum x_i}{n}
Sample Standard Deviation (s) = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
Margin of Error (ME) = t \cdot \left(\frac{s}{\sqrt{n}}\right)
Confidence Interval = \bar{x} \pm ME
Theorems
Central Limit Theorem
t-distribution properties
Suitable Grade Level
Grades 11-12
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