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.

Brain Volumes A sample of human brain volumes (cm3) is obtained from those listed in Data Set 8 “IQ and Brain Size” in Appendix B: 1027, 1029, 1034, 1070, 1079, 1079, 963, 1439

The sample of human brain volumes in centimeter cube from data set 8 “IQ and Brain size” in Appendix B.

The given sample data arrange in increasing order as:

963, 1027, 1029, 1034, 1070, 1079, 1079, 1439.

In the given data, the sample size is 8. Each value is the proportion of $\frac{1}{8}$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,$\text{n}=8$ , the cumulative left areas can be expressed as

$\frac{1}{16},\frac{3}{16},\frac{5}{16},\frac{7}{16},\frac{9}{16},\frac{11}{16},\frac{13}{16} \text{and} \frac{15}{16}$

The cumulative left areas expressed in decimal form are 0.0625, 0.1875, 0.3125, 0.4375, 0.5625, 0.6875, 0.8125, and 0.9375.

Refer to the standard normal distribution table, the z-score value corresponding to 0.0625, 0.1875, 0.3125, 0.4375, 0.5625, 0.6875, 0.8125 and 0.9375 for a left-tailed test is equal to -1.53, -0.89, -0.49, -0.16, 0.16, 0.49, 0.89, 1.53.

Pair thesample values of human brain volume with the corresponding z-score in the form of (x, y) as:

(963-1.53), (1027,-0.89),(1029,-0.49), (1034,-0.16), (1070,0.16),

(1079, 0.49), ( 1079, 0.89), and (1439,1.53)

Steps to draw a Normal quantile plot are as follows:

1. Make horizontal axis and vertical axis.
2. Mark the points 32, 33.3, 34.9 up to 45 on the horizontal axis and points -1.3, -1.04, -0.78, up to 1.3.
3. Provide title to horizontal and vertical axis as “X values” and “Z-score” respectively.
4. Mark the co-ordinates and obtain the graph as shown below.

From the normal quantile plot, the points do not show any symmetric pattern. So, the sample of human brain volume does not appear to be from a normally distributed population.