91Ó°ÊÓ

Assume that the population of \(x\) values has an approximately normal distribution. 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 \(6 \mathrm{mg} / \mathrm{dl}\) (Reference: Manual of Laboratory and Diagnostic Tests by F. Fischbach). Recently, the patient's total calcium tests gave the following readings (in \(\mathrm{mg} / \mathrm{d}\) l). $$ \begin{array}{rrrrrrr} 9.3 & 8.8 & 10.1 & 8.9 & 9.4 & 9.8 & 10.0 \\ 9.9 & 11.2 & 12.1 & & & & \end{array} $$ (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

Calculate the sample mean

Given in the problem, the sample mean is \( \bar{x} = 9.95 \). This is the average of the calcium readings provided.

02

Calculate the sample standard deviation

Also given, the sample standard deviation is \( s \approx 1.02 \). This measures the spread of the sample data points about the mean.

03

Determine the sample size

The sample consists of 11 values so the sample size \( n = 11 \).

04

Find the critical t-value

For a \(99.9\%\) confidence interval with \( n - 1 = 11 - 1 = 10 \) degrees of freedom, we use a t-table or calculator to find the critical t-value, which is approximately \( t^* = 4.587 \).

05

Calculate the margin of error (E)

The formula for the margin of error is \( E = t^* \times \frac{s}{\sqrt{n}} \). Plugging in the values, \( E = 4.587 \times \frac{1.02}{\sqrt{11}} \approx 1.41 \).

06

Construct the confidence interval

The \(99.9\%\) confidence interval is given by \( \bar{x} - E \) to \( \bar{x} + E \). Thus, the interval is \( 9.95 - 1.41 \) to \( 9.95 + 1.41 \), which is \( (8.54, 11.36) \).

07

Interpretation of the confidence interval

Since the average calcium level associated with tetany is below \(6 \ mg/dl\), and \(6\) is not within our confidence interval \((8.54, 11.36)\), it does not appear that this patient still has 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 is an essential concept in statistics, especially when dealing with data collected from a particular group or sample. Here, the sample mean is represented by \( \bar{x} \). It is calculated as the sum of all the data points divided by the number of data points. In this exercise, the calcium levels of the patient are the data points: 9.3, 8.8, 10.1, 8.9, 9.4, 9.8, 10.0, 9.9, 11.2, and 12.1 mg/dl. Adding these gives a total sum, and dividing by the count of values—11 in this case—results in a sample mean of 9.95 mg/dl.

This sample mean provides an estimate of the central tendency of the patient's calcium levels. It acts as a surrogate for the true population mean, offering insights into the average condition of the sample group. In this scenario, it helps us understand whether the patient's calcium levels reflect a deficiency.

T-Distribution

The t-distribution is a critical tool when working with small sample sizes or when the population standard deviation is unknown. It is particularly useful for constructing confidence intervals. The shape of the t-distribution is similar to the normal distribution but with thicker tails, which accounts for the increased variability inherent in small samples.

In this exercise, we use the t-distribution to find the critical t-value necessary for constructing a 99.9% confidence interval. Because the sample size is 11, we have 10 degrees of freedom \((n-1)\). A t-table or calculator tells us that the critical t-value for these parameters is approximately 4.587. This value stretches the interval further from the mean than a z-value would, reflecting the additional uncertainty presented by the smaller sample size.

Calcium Deficiency

Calcium deficiency, including conditions like tetany, is a medical issue where the calcium level in the blood is lower than what is considered healthy. An average calcium level below 6 mg/dl is associated with such deficiencies. It can lead to symptoms like severe muscle spasms, highlighting the importance of monitoring calcium levels in affected individuals.

In this case, a confidence interval is used to assess whether the patient still suffers from a calcium deficiency. The calculated interval spans from 8.54 to 11.36 mg/dl. This range lies entirely above the 6 mg/dl threshold, suggesting that the patient no longer experiences a calcium deficiency according to these readings.

Sample Standard Deviation

The sample standard deviation, denoted by \(s\), is a measure of how spread out the data points in a sample are around the sample mean. It gives an indication of the variability or dispersion within the sample data. A larger standard deviation means the data points are more spread out from the mean, while a smaller one indicates that they are closer to the mean.

In this problem, the sample standard deviation is approximately 1.02 mg/dl. This value is used in calculating the margin of error for the confidence interval, which in turn provides insight into the reliability of the sample mean as an estimate of the population mean. Understanding the standard deviation helps in gauging how much individual calcium readings differ from the average reading in the sample, thus aiding in making robust statistical inferences.