Q. 3.137 - Elementary Statistics

Question

 A study by researchers at the University of Maryland addressed the question of whether the mean body temperature of humans is $98.6^{°}F$ The results of the study by P. Mackowiak et al. appeared in the article "A Critical Appraisal of $98.6°F$ the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" (Journal of the American Medical Association ). Among other data, the researchers obtained the body temperatures of $93$healthy humans. The temperatures had a mean of $98.1°F$and a standard deviation of $0.65°F$

a. Construct a graph
b. Apply Chebyshev's rule with $k=2$ to make pertinent statements about the observations in the sample.
c. Repeat part (b) with $k=3$

Step-by-Step Solution

Answer

Part(a) Required graph is given below.

Part(b) At least $70$ healthy person have normal body temperature.

Part(c) At least $83$ healthy person have normal body temperature.

Part(a) Step 1 : Given information

We are given that sample has $93$ healthy humans

Mean $\overline{x}=98.1°F$

Standard deviation $S=0.65°F$

Part(a) Step 2 : Simplify

According to Chebyshev's Rule

$$\begin{gathered} \overline{x}=98.1°F \\ \overline{x}-s=97.45°F \\ \overline{x}-2s=96.8°F \\ \overline{x}-3s=96.15°F \\ \overline{x}+s=98.75°F \\ \overline{x}+2s=99.40°F \\ \overline{x}+3s=100.05°F \end{gathered}$$

Required graph is

Part(b) Step 1 : Given information

We are given that sample has $93$healthy humans

Mean $\overline{x}=98.1°F$

Standard deviation $s=0.65°F$

Part(b) Step 2 : Simplify

According to Chebyshev's rule ,if

$k=2$, then at least $75\%$ of the  person in the sample within two standard deviations to either side of the mean.

 Now,

$93\times 0.75\approx 70$

 Now, two standard deviations to either side of the mean is from $96.8°F$to $99.4°F$ 

Interpretation- At least $70$ out of $93$ have normal body temperature in the sample between $96.8°F$ and $99.4°F$

Part(c) Step 1 : Given information

We are given that sample has $93$ healthy humans.

Mean $\overline{x}=98.1°F$

Standard deviation $s=0.65°F$

Part(c) Step 2 : Simplify

According to Chebyshev's rule ,if

$k=3$, then at least $89\%$ of the adult females in the sample within two standard deviations to either side of the mean.

 Now

$93\times 0.89\approx 83$

 Now, two standard deviations to either side of the mean is from 

$97.90°F$ to $98.30°F$

Interpretation- At least $83$ out of $93$ have normal body temperature in the sample between $97.90°F$and$98.30°F$