Constructing Normal Quantile Plots. In Exercises 17–20, use the given data values to identify the corresponding z scores that are used for a normal quantile plot, then identify the coordinates of each point in the normal quantile plot. Construct the normal quantile plot, and then determine whether the data appear to be from a population with a normal distribution.

Female Arm Circumferences A sample of arm circumferences (cm) of females from Data Set 1 “Body Data” in Appendix B: 40.7, 44.3, 34.2, 32.5, 38.5.

The sample arm circumferences of females in centimeters from data set 1“Body Data” in Appendix B.

Arrangement of the given sample data in increasing order:

32.5, 34.2, 38.5, 40.7, 44.3

In the given data, the sample size is 5. Each value is the proportion of $\frac{1}{5}$of the sample.

So, the cumulative left areas can be expressed as

$\frac{1}{2n},\frac{3}{2n},\frac{5}{2n}, \text{and} \text{so} \text{on}$

For the given sample size, , the cumulative left areas, can be expressed as

$\frac{1}{10},\frac{3}{10},\frac{5}{10},\frac{7}{10} \text{and} \frac{9}{10}$

The cumulative left areas expressed in decimal form are 0.1000, 0.3000, 0.5000, 0.7000, and 0.9000.

Referring to the standard normal distribution table, the z-score values corresponding to cumulative left area of 0.1000, 0.3000, 0.5000, 0.7000, and 0.9000 is equal to -1.28, -0.52, 0, 0.52, and 1.28.

Pair thesample values of arm circumferences of females with the corresponding z-score in the form of (x, y) as:

(32.5, -1.28), (34.2, -0.52), (38.5, 0), (40.7, 0.52), (44.3, 1.28).

Steps to draw a Normal quantile plot are as follows:

1. Make horizontal axis and vertical axis.
2. Mark the coordinates on the graph with observed values on x-axis and z- scores on y-axis.
3. Provide title to horizontal and vertical axis as “X values” and “Z-score” respectively.
4. The normal quartile plot is shown below.

From the normal quantile plot, thepoints appear to lie reasonably close to a straight line. So, the sample of arm circumferences of female appears to be from a normally distributed population.