Question

the body - temperatures of a group of healthy adults have a bell - shaped distribution with a mean of 98.39°f and a standard deviation of 0.53°f. using the empirical rule, find each approximate percentage below.
a. what is the approximate percentage of healthy adults with body temperatures within 3 standard deviations of the mean, or between 96.80°f and 99.98°f?
b. what is the approximate percentage of healthy adults with body temperatures between 97.33°f and 99.45°f?
a. approximately % of healthy adults in this group have body temperatures within 3 standard deviations of the mean, or between 96.80°f and 99.98°f
(type an integer or a decimal. do not round.)
b. approximately % of healthy adults in this group have body temperatures between 97.33°f and 99.45°f
(type an integer or a decimal. do not round.)

Explanation:

Step1: Recall empirical rule for normal distribution

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

a.

The empirical rule for a normal distribution states that approximately 99.7% of the data lies within 3 standard deviations of the mean.
For part a:

1. For part a:

• Given mean $\mu = 98.39^{\circ}F$ and standard deviation $\sigma = 0.5^{\circ}F$.
• The range of values for a normal distribution is from $\mu - 3\sigma$ to $\mu+3\sigma$.
• The percentage of data within 3 standard deviations of the mean is approximately 99.7%.
• For a bell - shaped (normal) distribution, the proportion of data within 3 standard deviations of the mean is 99.7%.

Answer:

a. Approximately 99.7% of healthy adults have body temperatures within 3 standard deviations of the mean.
b. Approximately 95% of healthy adults have body temperatures within 2 standard deviations of the mean.
c. Approximately 68% of healthy adults have body temperatures within 1 standard deviation of the mean.

b.

1. First, find the number of standard deviations for the given temperature ranges:

• For part a:
• The mean $\mu = 98.39^{\circ}F$ and the standard deviation $\sigma = 0.5^{\circ}F$.
• To find the number of standard deviations for a given temperature range, we use the formula $z=\frac{x - \mu}{\sigma}$.
• For example, if $x_1 = 97.39^{\circ}F$ and $\mu = 98.39^{\circ}F$ and $\sigma = 0.5^{\circ}F$, then $z=\frac{97.39 - 98.39}{0.5}=-2$.
• For part b:
• We calculate the number of standard deviations for each value in a similar way.
• For part c:
• We continue this process for all values.
• For part d:
• And so on for all the values.
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• And so on for all the values.
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• And so on for all the values.
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