mework The body temperatures of a group of healthy adults have a bell-shaped distribution with a mean of \( 98.09^{\circ} \mathrm{F} \) and a standard deviation of \( 0.68^{\circ} \mathrm{F} \). Using the empirical rule, find each approximate percentage below. a. What is the approximate percentage of healthy adults with body temperatures within 2 standard deviations of the mean, or between \( 96.73^{\circ} \mathrm{F} \) and \( 99.45^{\circ} \mathrm{F} \) ? b. What is the approximate percentage of healthy adults with body temperatures between \( 97.41^{\circ} \mathrm{F} \) and \( 98.77^{\circ} \mathrm{F} \) ?
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In a bell-shaped distribution, the empirical rule states that about 95% of the data falls within 2 standard deviations of the mean. Since the mean body temperature is \( 98.09^{\circ} \mathrm{F} \) and the standard deviation is \( 0.68^{\circ} \mathrm{F} \), the range from \( 96.73^{\circ} \mathrm{F} \) to \( 99.45^{\circ} \mathrm{F} \) indeed captures approximately 95% of healthy adults. Now, for the second part, we’ll examine the temperatures \( 97.41^{\circ} \mathrm{F} \) and \( 98.77^{\circ} \mathrm{F} \). These correspond to roughly 1 to 2 standard deviations from the mean (as \( 97.41^{\circ} \mathrm{F} \) is about one standard deviation below the mean, and \( 98.77^{\circ} \mathrm{F} \) is about one standard deviation above). According to the empirical rule, around 68% of healthy adults fall within 1 standard deviation, which would apply here. So, approximately 68% of healthy adults will have body temperatures between \( 97.41^{\circ} \mathrm{F} \) and \( 98.77^{\circ} \mathrm{F} \).
