Q. 3.187 - Elementary Statistics

Question

Body Temperature. 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, Vol. data-custom-editor="chemistry" $268$, pp. $1578-1580$) Among other data, the researchers obtained the body temperatures of $93$ healthy humans, as provided on the Weiss Stats site

Step-by-Step Solution

Answer

(a) The quartiles are $97.65, 98.2, 98.6$

(b) The interquartile range is, $0.95$

(c) Five-number summary is, $96.7, 97.65, 98.2, 98.6, 99.4$

(d) The Potential outliers are, $49$

(e) The required boxplot is given below.

Part (a) Step 1: Given information

We are given that,

Sorted data is given in the Weiss stats which are as follows,

$$\begin{gathered} 96.7, 96.8, 96.9, 97, 97, 97, 97, 97.1, 97.2, 97.2, 97.2, 97.3, 97.3,97.4, \\ 97.4, 97.4, 97.4, 97.4, 97.4, 97.6, 97.6, 97.6, 97.6, 97.7, 97.7, 97.7, \\ 97.8, 97.8, 97.8, 97.8, 97.9, 98, 98, 98, 98, 98, 98, 98, 98, 98, 98, 98, \\ 98.1, 98.1, 98.1, 98.2, 98.2, 98.2, 98.2, 98.2, 98.2, 98.2, 98.2, 98.3, \\ 98.3, 98.4, 98.4, 98.4, 98.4, 98.4, 98.4, 98.5, 98.5, 98.5, 98.5, 98.6, \\ 98.6, 98.6, 98.6, 98.6, 98.6, 98.6, 98.6, 98.6, 98.7, 98.7, 98.8, 98.8, \\ 98.8, 98.8, 98.8, 98.8, 98.8, 99, 99, 99, 99, 99.1, 99.2, 99.2, 99.3, \\ 99.3, 99.4 \end{gathered}$$

Part (a) Step 2: Simplify

As we know that median is the middle value of a sorted data set. Since the number of data values is odd, the median is:-

    $Q_2=98.2$

So, roughly $50\%$ the states have percentage of adults in required condition is below $98.2° F$

Now, the first quartile is the median of values below $Q_2$ i.e.

       $Q_1=97.65$

So, roughly $25\%$ the states have percentage of adults in required condition is below $97.65° F$

Now,  the third quartile is the median of values below $Q_2$

      $Q_3=98.6$

So, roughly $75\%$ the states have percentage of adults in required condition is below $98.6° F$

Part (b) Step 1: Given information

We need to find out the interquartile range  

Part (b) Step 2: Simplify

The interquartile range is the difference between $Q_1$ and $Q_3$

     $IQR=Q_3-Q_1=98.6-97.65=0.95$

The middle $50\%$ of the healthy humans have body temperature varies by $0.95° F$

Part (c) Step 1: Given information

We need to find out the five-number summary  

Part (c) Step 2: Explanation

The five-number summary is minimum$=96.7$ , first quartile$=97.65$, second quartile$=98.2$ , third quartile$=98.6$ and maximum$=99.4$  

Part (d) Step 1: Given information

We need to find out the potential outliers.  

Part (d) Step 2: Simplify

An outlier is more than $1.5 IQR$ or greater than $Q_3$ or less than $Q_1$

Therefore,

        $$\begin{gathered} Q_3+\left(1.5 IQR\right)=98.6+\left(1.5\times 0.95\right)=100.025 \\ {Q}_3-\left(1.5 IQR\right)=97.65-\left(1.5\times 0.95\right)=97.175 \end{gathered}$$

Hence, $49$ is outlier because it is not in range of outliers

Part (e) Step 1: Given information

We need to find the boxplot which is given below  

Part(e) Step 2: Simplify

The whiskers of the boxplot are at a low and high value. The box starts at $Q_1$and ends at $Q_3$ and has a straight line at the median.