Question

pulse rates of women are normally distributed with a mean of 77.5 beats per minute and a standard deviation of 11.6 beats per minute. answer the following questions.
what are the values of the mean and standard deviation after converting all pulse rates of women to z scores using $z=\frac{(x-mu)}{sigma}$?
$mu = 0$
$sigma = 1$
the original pulse rates are measured with units of \beats per minute\. what are the units of the corresponding z scores? choose the correct choice below.
a. the z scores are measured with units of \beats per minute.\
b. the z scores are measured with units of \beats.\
c. the z scores are measured with units of \minutes per beat.\
d. the z scores are numbers without units of measurement.

Explanation:

Step1: Recall the properties of z - scores

The z - score formula is \(z=\frac{x - \mu}{\sigma}\). When we convert a set of data \(x\) (with mean \(\mu\) and standard deviation \(\sigma\)) to z - scores, the new mean of the z - scores is \(0\) and the new standard deviation is \(1\).

Step2: Analyze the units of z - scores

The formula \(z=\frac{x - \mu}{\sigma}\) involves subtracting the mean (\(\mu\)) from each data point \(x\) (so the units of \(x-\mu\) are the same as the units of \(x\)) and then dividing by the standard deviation (\(\sigma\) which has the same units as \(x\)). When we divide two quantities with the same units (e.g., if \(x\) has units of "beats per minute", then \(\mu\) has units of "beats per minute" and \(\sigma\) has units of "beats per minute"), the units cancel out.

Answer:

For the first question: The mean of the z - scores is \(0\) and the standard deviation of the z - scores is \(1\).
For the second question: D. The z scores are numbers without units of measurement.