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.

• 1.28, -0.52, 0, 0.52, 1.28

(round to two decimal places as needed. use ascending order.)
identify the coordinates of each point in the normal quantile plot. use ordered pairs of the form (x,y), where x is the sorted arm circumferences, and y is the corresponding z score.
(32.1, -1.28), (33.8, -0.52), (39.1, 0), (40.7, 0.52), (45,1.28)
(type ordered pairs. round to two decimal places as needed. use ascending order.)
construct the normal quantile plot.

Explanation:

Step1: Sort the data

First, sort the given data values \(32.1,45.0,39.1,40.7,33.8\) in ascending order. We get \(32.1,33.8,39.1,40.7,45.0\).

Step2: Determine the \(z -\)scores

For a sample of size \(n = 5\), the cumulative areas from the left for the data points (using the formula \(A=\frac{i - 0.5}{n}\), where \(i\) is the index of the data point in the sorted list) are:

• For \(i = 1\): \(A=\frac{1-0.5}{5}=0.1\). The \(z -\)score corresponding to \(A = 0.1\) (using the standard normal table) is \(z=-1.28\).
• For \(i = 2\): \(A=\frac{2 - 0.5}{5}=0.3\). The \(z -\)score corresponding to \(A = 0.3\) is \(z=-0.52\).
• For \(i = 3\): \(A=\frac{3-0.5}{5}=0.5\). The \(z -\)score corresponding to \(A = 0.5\) is \(z = 0\).
• For \(i = 4\): \(A=\frac{4-0.5}{5}=0.7\). The \(z -\)score corresponding to \(A = 0.7\) is \(z = 0.52\).
• For \(i = 5\): \(A=\frac{5-0.5}{5}=0.9\). The \(z -\)score corresponding to \(A = 0.9\) is \(z = 1.28\).

Step3: Form ordered pairs

The ordered pairs \((x,y)\) for the normal - quantile plot are formed by taking the sorted data values \(x\) and their corresponding \(z -\)scores \(y\). So the ordered pairs are \((32.1,-1.28),(33.8,-0.52),(39.1,0),(40.7,0.52),(45.0,1.28)\).

Step4: Check for normality

When we plot these points, if the points lie approximately on a straight line, the data is likely from a normal distribution. Looking at the ordered pairs \((32.1,-1.28),(33.8,-0.52),(39.1,0),(40.7,0.52),(45.0,1.28)\), the points lie approximately on a straight line.

Answer:

The data appear to be from a population with a normal distribution.