Body Temperature Data Set 5 “Body Temperatures” in Appendix B includes a sample of 106 body temperatures having a mean of and a standard deviation of 0.62F (for day 2 at 12 AM). Construct a 95% confidence interval estimate of the standard deviation of the body temperatures for the entire population.

Verified step by step guidance

1. Step 1: Understand the problem. We are tasked with constructing a 95% confidence interval for the population standard deviation (σ) based on a sample of 106 body temperatures. The sample standard deviation (s) is given as 0.62°F.
2. Step 2: Recall the formula for the confidence interval of the population standard deviation. The confidence interval is based on the chi-square distribution and is given by: \( \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{upper}}}} \leq \sigma \leq \sqrt{\frac{(n-1)s^2}{\chi^2_{\text{lower}}}} \), where \( n \) is the sample size, \( s \) is the sample standard deviation, and \( \chi^2_{\text{upper}} \) and \( \chi^2_{\text{lower}} \) are the critical values of the chi-square distribution for the given confidence level.
3. Step 3: Identify the degrees of freedom (df) and the critical values. The degrees of freedom are \( df = n - 1 \), where \( n \) is the sample size. For a 95% confidence level, find the critical values \( \chi^2_{\text{lower}} \) and \( \chi^2_{\text{upper}} \) from the chi-square distribution table or using statistical software.
4. Step 4: Plug the values into the formula. Substitute \( n = 106 \), \( s = 0.62 \), and the critical values \( \chi^2_{\text{lower}} \) and \( \chi^2_{\text{upper}} \) into the confidence interval formula to calculate the lower and upper bounds for \( \sigma \).
5. Step 5: Interpret the result. The resulting interval provides a range of plausible values for the population standard deviation of body temperatures at a 95% confidence level. This means we are 95% confident that the true population standard deviation lies within this interval.

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 expressed as a percentage. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population mean or standard deviation.

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 quantify how much individual body temperatures deviate from the average temperature.

Chi-Square Distribution

The Chi-square distribution is a statistical distribution that is commonly used in hypothesis testing and constructing confidence intervals for variance and standard deviation. When estimating the standard deviation of a population from a sample, the Chi-square distribution helps determine the critical values needed to construct the confidence interval, particularly when the sample size is small or the population variance is unknown.