Question

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

Short answer

The coordinates for the plot are:

Observed	Z-scores
963	-1.53
1027	-0.89
1029	-0.49
1034	-0.16
1070	0.16
1079	0.49
1079	0.89
1439	1.53

The normal quantile plot for given sample is:

The sample of human brain volumes does not appear to be from a population with a normal distribution.

Step-by-step solution

Given information

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

Arrange the given data in increasing order

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

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

Find the Cumulative left areas.

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.

Express the sample values and z-score in (x,y) coordinate .

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)

Make a Normal Quantile Plot  

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.

Conclude from Normal Quantile Plot

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.