Glucose concentrations in the blood above 110 mg / dL can be an early indication of several medi

step 1 For a given set of glucose measurements for a patient, we are to calculate the mean and standard

Glucose concentrations in the blood above 110 mg / dL can be...

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 the following 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?

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 the following 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?

Key Concepts

Mean

The mean is the arithmetic average of a set of numbers. It is calculated by summing all observations and dividing by the number of observations. This measure of central tendency provides a useful summary of the data set and is foundational for many statistical analyses.

Standard Deviation

Standard deviation quantifies the amount of variation or dispersion in a set of data values. A lower standard deviation indicates that the data points tend to be close to the mean, while a higher standard deviation shows that the data points are spread out over a larger range of values. This concept is crucial for understanding the variability within a sample.

Confidence Interval

A confidence interval is a range of values, derived from the sample statistics, that is likely to contain the true parameter of the population (for example, the mean) with a specified level of confidence (commonly 95%). It provides an estimate of uncertainty around the sample mean that results from sampling variability.

t-Distribution

When dealing with small sample sizes or when the population standard deviation is unknown, the t-distribution is used instead of the normal distribution. It accounts for additional uncertainty by having heavier tails, and it is characterized by degrees of freedom that are related to the sample size. This adjustment allows for more accurate confidence intervals for the mean in small-sample contexts.

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