Problem 74 - Chemistry The Science in Context

Question

Diabetes Test Glucose concentrations in the blood above \(110 \mathrm{mg} / \mathrm{dL}\) can be an early indication of several medical conditions, including diabetes. Suppose analyses of a series of blood samples from a patient at risk of diabetes produce these results: \(106,99,109,108,\) and \(105 \mathrm{mg} / \mathrm{dL}\) a. What are the mean and the standard deviation of the data? b. Patients with blood glucose levels above \(120 \mathrm{mg} / \mathrm{dL}\) are considered diabetic. Is this value within the \(95 \%\) confidence interval of these data?

Step-by-Step Solution

Answer

Instructions: Provide the following steps in your answer: 1) Calculate the mean of the glucose concentrations 2) Calculate the standard deviation 3) Calculate the standard error 4) Determine the 95% confidence interval 5) Check if 120 mg/dL is within this confidence interval.

Step 1: Calculate the mean of the glucose concentrations

To calculate the mean, add all the glucose concentrations together and divide by the total number of samples. The formula is: \(\mathrm{Mean} = \frac{\mathrm{Sum\: of\: glucose\: levels}}{\mathrm{Total\: number\: of\: samples}}\)

Step 2: Calculate the standard deviation

To calculate the standard deviation, first find the squared differences between each glucose level and the mean, then find the average of those squared differences, and finally take the square root of that average. The formula is: \(\mathrm{Standard\: Deviation} = \sqrt{\frac{\mathrm{Sum\: of\: squared\: differences}}{\mathrm{Total\: number\: of\: samples}}}\) b. Checking the 95% confidence interval:

Step 3: Calculate the standard error

Standard error is the measure of how spread out the glucose levels are from the mean. The formula for standard error is: \(\mathrm{Standard\: Error} = \frac{\mathrm{Standard\: Deviation}}{\sqrt{\mathrm{Total\: number\: of\: samples}}}\)

Step 4: Determine the 95% confidence interval

Assuming the data has a normal distribution, 95% of data lies within two standard errors of the mean. The formula for the confidence interval is: \(\mathrm{Mean} \pm 2 \times \mathrm{Standard\: Error}\)

Step 5: Check if 120 mg/dL is within the confidence interval

Compare the upper limit of the confidence interval to the value of 120 mg/dL. If the value 120 mg/dL falls within the interval, we may consider it as a possible normal blood glucose level based on these data.

Key Concepts

Diabetes Test

Diabetes is a condition that affects how the body processes blood glucose, the primary source of energy. A diabetes test usually measures blood glucose levels to determine a person's risk or presence of diabetes.
Blood glucose concentrations above certain thresholds can indicate potential health issues. For instance, levels over 110 mg/dL may suggest an increased risk for diabetes or other medical conditions.
In this context, analyzing blood samples to monitor glucose concentrations is crucial for early detection and management.

• Blood glucose concentrations are measured in milligrams per deciliter (mg/dL).
• It's important to measure at different times to understand fluctuations.
• Patients with persistent higher levels may require further testing.

This type of analysis can prove to be a useful tool for healthcare providers and patients aiming for early intervention and preventive care.

Mean Calculation

The mean, often referred to as the average, is a measure that summarizes a set of numbers. It gives an idea of the central tendency, or typical value, of the data set.
To find the mean of blood glucose levels, you add all the values together and divide by the number of values. This provides a single number that represents a typical glucose level from the sampled data.
Here is the step-by-step process to calculate the mean:

• Add all the glucose concentrations from the samples: 106 + 99 + 109 + 108 + 105.
• Divide the sum by the number of samples, in this case, 5.
• The result is the mean glucose level.

Using the mean helps to understand where the glucose levels of the patient tend to hover. This can help healthcare professionals make decisions regarding further testing or intervention.

Standard Deviation

Standard deviation is a measure of how much variation or spread there is from the average (mean). In a data set, a smaller standard deviation means the values are close to the mean, whereas a larger standard deviation indicates that the values are more spread out.
To compute the standard deviation for blood glucose levels, follow these steps:

• Calculate the difference between each glucose level and the mean.
• Square each difference to eliminate negative values.
• Take the average of these squared differences.
• Finally, take the square root of this average value.

Understanding standard deviation is essential because it sheds light on the consistency of glucose levels in a patient.
A larger variation might indicate irregular glucose levels, which can be a sign of an underlying issue that requires medical attention.

Confidence Interval

A confidence interval provides a range within which we expect the true mean of the population to lie, given our sample data. In the context of blood glucose levels, it helps estimate the range in which a patient's true average glucose level might fall.
To find the 95% confidence interval, we use the mean and standard deviation to calculate the standard error, then apply it to construct the interval:

• The standard error is calculated by dividing the standard deviation by the square root of the number of samples.
• The confidence interval is then determined by adding and subtracting twice the standard error from the mean.

The reason we use a 95% confidence interval is that it covers most variations in the sample and provides a high level of certainty.
In practical terms, if the 120 mg/dL threshold for diabetes lies outside this interval, the initial sample values might not support a diagnosis of diabetes with high confidence. This interval assists healthcare professionals in making informed decisions about a patient's health status.