In: Statistics and Probability

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 \(\mathrm{z}=\frac{(\mathrm{x}-\mu)}{\sigma}\) ?

The original pulse rates are measure with units of "beats per minute". What are the units of the corresponding z scores? Choose the correct choice below.

- The z scores are measured with units of "beats per minute".
- The z scores are measured with units of "minutes per beat".
- The z scores are measured with units of "beats."
- The z scores are numbers without units of measurement.

It is given that the pulse rates of women have mean values as 77.5, and the standard deviation is 11.6. All pulse rates are converted into z-scores by subtracting the mean from each of the values and dividing the corresponding result with the standard deviation.

The converted z-scores are as shown below:

$$ \begin{array}{|l|l|l|l|} \hline P U L S E & z-\text {scores} & P U L S E & z-\text {scores} \\ \hline 78 & 0.0431 & 90 & 1.07759 \\ \hline 80 & 0.21552 & 90 & 1.07759 \\ \hline 68 & -0.819 & 68 & -0.819 \\ \hline 56 & -1.8534 & 72 & -0.4741 \\ \hline 76 & -0.1293 & 82 & 0.38793 \\ \hline 78 & 0.0431 & 72 & -0.4741 \\ \hline 78 & 0.0431 & 78 & 0.0431 \\ \hline 90 & 1.07759 & 104 & 2.28448 \\ \hline 96 & 1.59483 & 62 & -1.3362 \\ \hline 60 & -1.5086 & 72 & -0.4741 \\ \hline 98 & 1.76724 & 72 & -0.4741 \\ \hline 66 & -0.9914 & 88 & 0.90517 \\ \hline 100 & 1.93966 & 74 & -0.3017 \\ \hline 76 & -0.1293 & 72 & -0.4741 \\ \hline 64 & -1.1638 & 82 & 0.38793 \\ \hline 82 & 0.38793 & 78 & 0.0431 \\ \hline 62 & -1.3362 & 78 & 0.0431 \\ \hline 72 & -0.4741 & 98 & 1.76724 \\ \hline 74 & -0.3017 & 64 & -1.1638 \\ \hline \end{array} $$

Using MINITAB, the descriptive statistic for the z-scores is as shown below:

From the MINITAB output, \(\mu=0\) and \(\sigma=0.996 \approx 1\)

Here, the pulse rates are measured with given units. But the corresponding z-scores are the numbers or constant. So, the z-scores will not have any units of measurements.

Therefore, the correct option is D.

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