Mean Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of 98.20 F and a standard deviation of 0.62 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 F is the mean body temperature?

Verified step by step guidance

1. Step 1: Identify the given values from the problem. The sample size (n) is 106, the sample mean (x̄) is 98.20°F, the sample standard deviation (s) is 0.62°F, and the confidence level is 95%.
2. Step 2: Determine the critical value (t*) for a 95% confidence level. Since the sample size is greater than 30, we can use the t-distribution. The degrees of freedom (df) is calculated as n - 1, which is 106 - 1 = 105. Use a t-table or statistical software to find the t* value corresponding to a 95% confidence level and df = 105.
3. Step 3: Calculate the standard error of the mean (SE). The formula for SE is: ${\mathrm{SE}}_{\mathrm{x̄}}=\frac{s}{\sqrt{n}}$, where s is the sample standard deviation and n is the sample size.
4. Step 4: Compute the margin of error (ME). The formula for ME is: $\mathrm{ME}=\mathrm{t*}\times {\mathrm{SE}}_{\mathrm{x̄}}$, where t* is the critical value and SE is the standard error of the mean.
5. Step 5: Construct the confidence interval. The formula for the confidence interval is: $\mathrm{x̄}\pm \mathrm{ME}$. Substitute the sample mean and the margin of error to find the lower and upper bounds of the confidence interval. Finally, interpret the result in the context of the problem, comparing the interval to the commonly believed mean of 98.6°F.

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a data set, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. It is calculated using the sample mean, standard deviation, and the appropriate z or t value based on the sample size. This interval provides insight into the precision of the sample estimate and helps assess the reliability of the mean.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range. In the context of the body temperature data, it helps to understand how much individual body temperatures deviate from the mean temperature of 98.20 F.

Population Mean vs. Sample Mean

The population mean is the average of a set of values for an entire population, while the sample mean is the average calculated from a subset of that population. In this case, the sample mean of 98.20 F is used to estimate the population mean. Understanding the difference is crucial for interpreting the results of the confidence interval and assessing the validity of the common belief that the average body temperature is 98.6 F.