Question

a personal trainer was completing some research on the resting heart rates of people who exercise regularly. as part of his research, he randomly selected 75 people who exercise regularly and calculated the standard deviation for their resting heart rates to be 7 beats per minute (bpm). identify the correct formula and/or calculator function that would be used to construct a confidence interval for the standard deviation of resting heart rates (in bpm) for those who exercise regularly. formula: $lb = \frac{(n - 1)s^2}{\chi^2_{\frac{\alpha}{2}}}$ and $ub = \frac{(n - 1)s^2}{\chi^2_{1 - \frac{\alpha}{2}}}$ formula: $\hat{p} \pm z_{\frac{\alpha}{2}} \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$ calculator function: 1 - propzint formula: $\bar{x} \pm t_{\frac{\alpha}{2}} \cdot \frac{s}{\sqrt{n}}$ calculator function: t - interval formula: $lb = \sqrt{\frac{(n - 1)s^2}{\chi^2_{\frac{\alpha}{2}}}}$ and $ub = \sqrt{\frac{(n - 1)s^2}{\chi^2_{1 - \frac{\alpha}{2}}}}$

Explanation:

Step1: Identify the parameter of interest

The problem asks for a confidence interval for the standard deviation of resting heart rates.

Step2: Match to the correct method

Confidence intervals for population standard deviation use the chi-square distribution. The formula for the confidence interval of the standard deviation is the square root of the chi-square interval for variance:
Lower bound (LB) = $\sqrt{\frac{(n-1)s^2}{\chi^2_{\alpha/2}}}$
Upper bound (UB) = $\sqrt{\frac{(n-1)s^2}{\chi^2_{1-\alpha/2}}}$

Answer:

Formula: $LB=\sqrt{\frac{(n - 1)s^2}{\chi^2_{\frac{\alpha}{2}}}}$ and $UB=\sqrt{\frac{(n - 1)s^2}{\chi^2_{1-\frac{\alpha}{2}}}}$ (the last option)