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Diagnostic Tests: Total Calcium 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 (Reference: Manual of Laboratory and Diagnostic Tests by F. Fischbach). Recently, the patient's total calcium tests gave the following readings (in mg/dl). $$ \begin{array}{ccccccc} 9.3 & 8.8 & 10.1 & 8.9 & 9.4 & 9.8 & 10.0 \\ 9.9 & 11.2 & 12.1 & & & & \end{array} $$ (a) Use a calculator to verify that \(\bar{x}=9.95\) and \(s \approx 1.02\) (b) Find a \(99.9 \%\) confidence interval for the population mean of total calcium in this patient's blood. (c) Interpretation Based on your results in part (b), does it seem that this patient still has a calcium deficiency? Explain.

Step by step solution

01

Verify the Sample Mean

We are given the sample of total calcium readings: \(9.3, 8.8, 10.1, 8.9, 9.4, 9.8, 10.0, 9.9, 11.2, 12.1\). Calculate the sample mean \(\bar{x}\) by adding all readings together and dividing by the number of readings, which is 10.\[ \bar{x} = \frac{9.3 + 8.8 + 10.1 + 8.9 + 9.4 + 9.8 + 10.0 + 9.9 + 11.2 + 12.1}{10} = 9.95 \]The given \(\bar{x} = 9.95\) is verified.

02

Verify the Sample Standard Deviation

The sample standard deviation \(s\) is calculated using the formula:\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \]Substitute the values in:\[ s \approx \sqrt{\frac{(9.3 - 9.95)^2 + (8.8 - 9.95)^2 + \ldots + (12.1 - 9.95)^2}{10 - 1}} \approx 1.02 \]The given \(s \approx 1.02\) is verified.

03

Find the Critical Value for 99.9% Confidence Interval

For a 99.9% confidence interval and given the sample size of 10, we use a t-distribution with \(n-1=9\) degrees of freedom. Look up the critical t-value \(t^*\) for 99.9% confidence level, which is \(t^* \approx 4.297\).

04

Calculate the Confidence Interval

Use the formula for the confidence interval:\[ \bar{x} \pm t^* \cdot \frac{s}{\sqrt{n}} \]Substituting the values:\[ 9.95 \pm 4.297 \cdot \frac{1.02}{\sqrt{10}} \approx 9.95 \pm 1.384 \]This gives us the confidence interval \([8.566, 11.334]\).

05

Interpretation

The confidence interval \([8.566, 11.334]\) for the population mean indicates that the mean total calcium level is above 6 mg/dl. Since this interval does not include values below 6 mg/dl, it suggests that the patient does not currently have a calcium deficiency.

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Key Concepts

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

Sample Mean

The Sample Mean, denoted as \( \bar{x} \), is a fundamental concept when working with statistics. It represents the average value of a set of numbers. In our calcium example, we have 10 different readings, and our task is to find out the mean of these readings. If you sum up all the readings: 9.3, 8.8, 10.1, 8.9, 9.4, 9.8, 10.0, 9.9, 11.2, 12.1, you get a total of 99.5 mg/dl. The sample mean is then calculated by dividing this total by the number of observations, which is 10. Therefore, the sample mean \( \bar{x} \) equals \( 9.95 \) mg/dl. This measure tells us the average calcium level from our set of readings, giving us a single value to represent the data set.

• Measurement: The sample mean provides a measure that summarizes all the data points.
• Importance: It serves as a central tendency measure, indicating where most of our data points cluster.

Standard Deviation

The Standard Deviation, denoted as \( s \), is another critical statistical tool used to understand how spread out the data points in your data set are. It measures the variability or dispersion. In simpler terms, it gives us the average distance of each data point from the mean. To find the sample standard deviation in our calcium example, we use the formula:\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \] Substituting the calcium readings into this formula, we evaluate the squared differences from the mean, sum them, and divide by \( n-1 \) (which is 9 as there are 10 observations). The result is then square rooted, which comes out approximately to \( 1.02 \).

• Utility: It informs us about the spread of our data points. A smaller standard deviation indicates that the data points are close to the mean, while a larger one suggests they are more spread out.
• Relevance: Understanding the variation in calcium readings can help determine whether observations are consistent or if there are unusual spikes or drops.

t-Distribution

The t-Distribution is particularly useful when dealing with small sample sizes, typically under 30, and when the population standard deviation is unknown. When creating confidence intervals for small samples, like our ten calcium readings, we rely on the t-distribution rather than the normal distribution.In our example, to calculate a 99.9% confidence interval for the mean total calcium level, we use a t-distribution with \( n-1 \) degrees of freedom (which is 9 here). The critical t-value for this confidence level and degrees of freedom is approximately \( 4.297 \). This value tells us how many standard deviations we can expect the sample mean to fall from the true population mean for our given level of confidence.

• Significance: The t-value helps in determining the range of our confidence interval, crucial for making inferences about the population mean.
• Application: In medical tests like our calcium levels, a correct understanding of this distribution allows us to deduce conclusions with the right amount of confidence.

Clinical Interpretation

Clinical Interpretation ties the statistical results back to their real-world implications, crucial in fields like medicine. After calculating a 99.9% confidence interval for our calcium data, we found it to be \([8.566, 11.334]\). These results are pivotal in assessing the patient's health.Since the range of this confidence interval is entirely above 6 mg/dl (the level indicating calcium deficiency), it suggests that the patient's average calcium level is above the deficiency threshold. Therefore, statistically, it indicates that the patient likely does not currently suffer from calcium deficiency.

• In Practice: Such interpretations help clinicians make informed decisions about patient treatment and health status.
• Communication: Ensuring patients understand that the statistical analysis supports their improved health status or the effect of their treatment is vital.