91Ó°ÊÓ

Refer to Exercise 1 for the data of body temperatures recorded from 20 subjects measured in degrees Fahrenheit.

The corresponding frequency distribution is as follows.

A histogram is a plot that depicts the frequencies of class intervals derived from continuous data using vertical bars. The length of the bars of a histogram shows the frequency corresponding to different intervals.

The population from which the data set is recorded is considered the normal population if the histogram depicts the following characteristics.

• It should be bell-shaped. That is, the frequencies should start from low, then attain a peak, and then gradually decrease.
• It should be approximately symmetric. That is, the two halves of the graph should be mirror images of each other.

From Exercise 1, the frequency distribution is obtained as follows.

The midpoint of any class interval is computed using the following formula.

${\text{Midpoint}}_i=\frac{L{L}_i+U{L}_i}{2}$

Here,$L{L}_i,U{L}_i$

are the lower and upper limits of the ith class interval.

The midpoints of the class intervals are computed as below.

$$\begin{gathered} {\text{Midpoint}}_1=\frac{97.0+97.4}{2} \\ =97.2 \\ {\text{Midpoint}}_2=\frac{97.5+97.9}{2} \\ =97.7 \\ {\text{Midpoint}}_3=\frac{98.0+98.4}{2} \\ =98.2 \\ {\text{Midpoint}}_4=\frac{98.5+98.9}{2} \\ =98.7 \\ {\text{Midpoint}}_5=\frac{99.0+99.4}{2} \\ =99.2 \end{gathered}$$

The midpoints corresponding to the class intervals are tabulated as follows.

Use the given steps to construct the histogram.

• Sketch the histogram for the midpoints and frequencies of the histogram.
• Mark the values from 97.2 up to 99.2 with a gap of 0.5 units between each observation on the horizontal axis.
• Mark the values from 0 to 8 with a gap of 1 unit between any two observations on the vertical axis.
• Draw vertical bars of even width with lengths equal to the frequency corresponding to each midpoint.
• Label the horizontal axis as ‘Body Temperature’ and the vertical axis as ‘Frequency’.

The histogram is constructed as follows.

It can be observed that the frequencies start low, then increase to reach the maximum, and then decrease eventually.

Thus, the histogram is bell-shaped.

Also, the two halves of the histogram are approximately mirrored images of each other.

Thus, the histogram is symmetric.

Therefore, it can be concluded that the data comes from a normal population.