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 quartile plot, then identify the coordinates of each point in the normal quartile plot. construct the normal quartile 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 quartile 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 quartile plot. use ordered pairs of the form (x,y), where x is the sorted arm circumferences, and y is the corresponding z score.
32.10, -1.28, 33.80, -0.52, 39.10, 0.00, 40.70, 0.52, 45.00, 1.28
(type ordered pairs. round to two decimal places as needed. use ascending order.)

Explanation:

Step1: Sort the data

First, sort the sample of female arm circumferences: \(32.1,33.8,39.1,40.7,45.0\)

Step2: Find the \(z -\)scores

For \(n = 5\) data points, the cumulative areas for the left - tails are calculated as follows:
The formula for the cumulative area \(A=\frac{i - 0.5}{n}\), where \(i = 1,2,\cdots,n\)

• When \(i = 1\), \(A=\frac{1 - 0.5}{5}=0.1\). Using the standard normal table (or a calculator with a normal distribution function), the \(z -\)score corresponding to \(A = 0.1\) is \(z=-1.28\)
• When \(i = 2\), \(A=\frac{2 - 0.5}{5}=0.3\). The \(z -\)score corresponding to \(A = 0.3\) is \(z=-0.52\)
• When \(i = 3\), \(A=\frac{3 - 0.5}{5}=0.5\). The \(z -\)score corresponding to \(A = 0.5\) is \(z = 0\)
• When \(i = 4\), \(A=\frac{4 - 0.5}{5}=0.7\). The \(z -\)score corresponding to \(A = 0.7\) is \(z = 0.52\)
• When \(i = 5\), \(A=\frac{5 - 0.5}{5}=0.9\). 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 pairing the sorted data \(x\) (arm circumferences) with the corresponding \(z -\)scores \(y\)

• For \(x = 32.1\) and \(y=-1.28\), the pair is \((32.1,-1.28)\)
• For \(x = 33.8\) and \(y=-0.52\), the pair is \((33.8,-0.52)\)
• For \(x = 39.1\) and \(y = 0\), the pair is \((39.1,0)\)
• For \(x = 40.7\) and \(y = 0.52\), the pair is \((40.7,0.52)\)
• For \(x = 45.0\) and \(y = 1.28\), the pair is \((45.0,1.28)\)

Step4: Check for normality

When constructing the normal - quantile plot, if the points approximately lie on a straight line, the data is likely from a population with a normal distribution. In this case, the points \((32.1,-1.28),(33.8,-0.52),(39.1,0),(40.7,0.52),(45.0,1.28)\) would form a pattern that is close to a straight line.

Answer:

The \(z -\)scores are \(-1.28,-0.52,0,0.52,1.28\). The ordered pairs for the normal - quantile plot are \((32.1,-1.28),(33.8,-0.52),(39.1,0),(40.7,0.52),(45.0,1.28)\). The data appears to be from a population with a normal distribution.