A simple random sample from a population with a normal distribution of 106 body temperatures has $\overline{x}={98.80}^{\circ }\mathrm{F}$ and $\mathrm{s}={0.67}^{\circ }\mathrm{F}$. Construct an $80\mathrm{\%}$ confidence interval estimate of the standard deviation of body temperature of all healthy humans.
Click the icon to view the table of Chi-Square critical values.
(Round to two decimal places as needed.)

Step 1 ：We are given a simple random sample from a population with a normal distribution of 106 body temperatures. The sample mean is $\overline{x}={98.80}^{\circ }F$ and the sample standard deviation is $s={0.67}^{\circ }F$. We are asked to construct an 80% confidence interval estimate of the standard deviation of body temperature of all healthy humans.

Step 2 ：To construct a confidence interval for the standard deviation, we can use the chi-square distribution. The formula for the confidence interval is given by: $$\sqrt{\frac{(n-1)s^2}{{\chi }_{\alpha /2,n-1}^2}},\sqrt{\frac{(n-1)s^2}{{\chi }_{1-\alpha /2,n-1}^2}}$$ where: n is the sample size, s is the sample standard deviation, and ${\chi }_{\alpha /2,n-1}^2$ and ${\chi }_{1-\alpha /2,n-1}^2$ are the chi-square critical values for $\alpha /2$ and $1-\alpha /2$ degrees of freedom respectively.

Step 3 ：In this case, we have n = 106, s = 0.67, and we want an 80% confidence interval, so $\alpha =1-0.80=0.20$. We need to find the chi-square critical values for $\alpha /2=0.10$ and $1-\alpha /2=0.90$ with n-1 = 105 degrees of freedom.

Step 4 ：Using the chi-square table, we find that the chi-square critical values are ${\chi }_{0.10,105}^2=86.91$ and ${\chi }_{0.90,105}^2=123.95$.

Step 5 ：Substituting these values into the formula, we find the lower and upper bounds of the confidence interval to be $\sqrt{\frac{(106-1)(0.67{)}^2}{86.91}}=0.62$ and $\sqrt{\frac{(106-1)(0.67{)}^2}{123.95}}=0.74$ respectively.

Step 6 ：Thus, the 80% confidence interval estimate of the standard deviation of body temperature of all healthy humans is $\boxed{{\displaystyle (0.62,0.74)}}$.