Question

use the given data values (a sample of female arm circumferences in centimeters) 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, then determine whether the data appear to be from a population with a normal distribution.
32.1, 45.0, 39.1, 40.7, 33.8
list the z scores for the normal quantile plot.
(round to two decimal places as needed. use ascending order.)

Explanation:

Step1: Sort the data

First, sort the data values in ascending order: \(32.1,33.8,39.1,40.7,45.0\)

Step2: Calculate the cumulative area

For \(n = 5\) data points, the cumulative area \(A_i=\frac{i - 0.5}{n}\), where \(i = 1,2,\cdots,n\)

• When \(i = 1\), \(A_1=\frac{1 - 0.5}{5}=0.1\)
• When \(i = 2\), \(A_2=\frac{2 - 0.5}{5}=0.3\)
• When \(i = 3\), \(A_3=\frac{3 - 0.5}{5}=0.5\)
• When \(i = 4\), \(A_4=\frac{4 - 0.5}{5}=0.7\)
• When \(i = 5\), \(A_5=\frac{5 - 0.5}{5}=0.9\)

Step3: Find the z - scores

Using the standard normal distribution table or a calculator with a normal - distribution function (e.g., invNorm on a TI - 84 Plus), we find the \(z\) - scores corresponding to the cumulative areas:

• For \(A = 0.1\), \(z_1\approx - 1.28\)
• For \(A = 0.3\), \(z_2\approx - 0.52\)
• For \(A = 0.5\), \(z_3 = 0\)
• For \(A = 0.7\), \(z_4\approx0.52\)
• For \(A = 0.9\), \(z_5\approx1.28\)

Answer:

\(-1.28,-0.52,0,0.52,1.28\)