Question

the body temperatures of a group of healthy adults have a bell - shaped distribution with a mean of ( 98.21^{circ}f ) and a standard deviation of ( 0.65^{circ}f ). using the empirical rule, find each approximate percentage below.
a. what is the approximate percentage of healthy adults with body temperatures within 1 standard deviation of the mean, or between ( 97.56^{circ}f ) and ( 98.86^{circ}f )?
b. what is the approximate percentage of healthy adults with body temperatures between ( 96.26^{circ}f ) and ( 100.16^{circ}f )?
a. approximately 68% of healthy adults in this group have body temperatures within 1 standard deviation of the mean, or between ( 97.56^{circ}f ) and ( 98.86^{circ}f ).
(typo an integer or a decimal. do not round)
b. approximately (square%) of healthy adults in this group have body temperatures between ( 96.26^{circ}f ) and ( 100.16^{circ}f )
(typo an integer or a decimal. do not round)

Explanation:

Step1: Recall the empirical rule

The empirical rule for a bell - shaped (normal) distribution states that approximately 68% of the data lies within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean.

Step2: Calculate the number of standard deviations for part b

The mean is \(\mu = 98.21^{\circ}F\) and the standard deviation is \(\sigma=0.65^{\circ}F\).
For \(x_1 = 96.26^{\circ}F\), the number of standard deviations from the mean is \(z_1=\frac{\mu - x_1}{\sigma}=\frac{98.21 - 96.26}{0.65}=\frac{1.95}{0.65}=3\)
For \(x_2 = 100.16^{\circ}F\), the number of standard deviations from the mean is \(z_2=\frac{x_2-\mu}{\sigma}=\frac{100.16 - 98.21}{0.65}=\frac{1.95}{0.65}=3\)

Answer:

a. Approximately \(68\%\) of healthy adults in this group have body temperatures within 1 standard deviation of the mean, or between \(97.56^{\circ}F\) and \(98.86^{\circ}F\).
b. Approximately \(99.7\%\) of healthy adults in this group have body temperatures between \(96.26^{\circ}F\) and \(100.16^{\circ}F\)