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.

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.

Short answer

The coordinates used for normal probability plot are:

Observations	Z-scores
32.5	-1.28
34.2	-0.52
38.5	0
40.7	0.52
44.3	1.28

The normal quantile plot for given sample is:

The arm circumferences of females appear to be from a population with a normal distribution.

Step-by-step solution

Given information

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

Arrange the given data in increasing order

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

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

Find the Cumulative left areas.

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.

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

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).

Sketch a Normal Quantile Plot  

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.

Conclude from Normal Quantile Plot

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.