Data Set 3 "Body Temperatures" in Appendix B includes a...

Data Set 3 "Body Temperatures" in Appendix B includes a sample of 106 body temperatures having a mean of $98.20^{\circ} \mathrm{F}$ and a standard deviation of $0.62^{\circ} \mathrm{F}$. Construct a $95 \%$ confidence interval estimate of the mean body temperature for the entire population. What does the result suggest about the common belief that $98.6^{\circ} \mathrm{F}$ is the mean body temperature?

Key Concepts

Confidence Interval

A confidence interval estimates a range within which a population parameter, such as a mean, is likely to lie based on sample data. It is constructed using the sample mean, variability (such as the standard deviation), and a multiplier that corresponds to the desired level of confidence, thereby providing a measure of uncertainty around the estimate.

Standard Error

The standard error quantifies the variability of a sample statistic, typically the mean, from the true population parameter. It is calculated as the sample standard deviation divided by the square root of the sample size, and it is crucial for determining the width of confidence intervals.

t-Distribution

When estimating a population mean from a sample where the population standard deviation is unknown and the sample size is relatively small, the t-distribution is used instead of the normal distribution. This distribution accounts for extra uncertainty by having thicker tails, which results in wider confidence intervals.

Interpretation of Confidence Intervals

This concept involves understanding that a confidence interval, such as a 95% interval, is not a probability statement about the parameter but indicates that, under repeated sampling, approximately 95% of such constructed intervals would contain the true population mean. It also allows for assessing claims, for example, comparing a common belief about a mean value to the estimated confidence range.

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