Question

Histogram of Body Temperatures Construct the histogram that corresponds to the frequency distribution from Exercise 1. Use the class midpoint values for the horizontal scale. Does the histogram suggest that the data are from a population having a normal distribution? Why or why not?

Short answer

The following histogram is constructed for the given data on body temperatures.

Yes, the data appears to be from a normal population as the histogram is bell-shaped, and it is approximately symmetric.

Step-by-step solution

Given information

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.

Body Temperatures	Frequency
97.0-97.4	2
97.5-97.9	4
98.0-98.4	7
98.5-98.9	5
99.0-99.4	2

Define a histogram

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.

Obtain the midpoints from the frequency distribution

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

Body Temperatures	Frequency
97.0-97.4	2
97.5-97.9	4
98.0-98.4	7
98.5-98.9	5
99.0-99.4	2

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.

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

Sketch the histogram

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

Body Temperatures	Frequency	Midpoints
97.0-97.4	2	97.2
97.5-97.9	4	97.7
98.0-98.4	7	98.2
98.5-98.9	5	98.7
99.0-99.4	2	99.2

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.

Analyse the histogram

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.