Questions: 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=(x-μ)/σ ? μ=0 σ=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." B. The z scores are measured with units of "minutes per beat." C. The z scores are measured with units of "beats per minute." D. The z scores are numbers without units of measurement.

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=(x-μ)/σ ? μ=0 σ=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." B. The z scores are measured with units of "minutes per beat." C. The z scores are measured with units of "beats per minute." D. The z scores are numbers without units of measurement.
Transcript text: 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 $z=\frac{(x-\mu)}{\sigma}$ ? \[ \begin{array}{l} \mu=0 \\ \sigma=1 \end{array} \] 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." B. The $z$ scores are measured with units of "minutes per beat." C. The $z$ scores are measured with units of "beats per minute." D. The $z$ scores are numbers without units of measurement.
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Solution

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Solution Steps

Step 1: Mean and Standard Deviation of Z-scores

When converting pulse rates to \( z \)-scores using the formula

\[ z = \frac{(x - \mu)}{\sigma} \]

the mean (\( \mu \)) of the \( z \)-scores becomes \( 0 \) and the standard deviation (\( \sigma \)) becomes \( 1 \). Therefore, we have:

\[ \text{Mean of } z\text{-scores} = 0 \] \[ \text{Standard deviation of } z\text{-scores} = 1 \]

Step 2: Units of Z-scores

The \( z \)-scores are dimensionless quantities, meaning they do not have any units of measurement. This is because \( z \)-scores represent the number of standard deviations a data point is from the mean, which cancels out the units. Thus, we conclude that:

\[ \text{The } z \text{ scores are numbers without units of measurement.} \]

Final Answer

The mean of the \( z \)-scores is \( 0 \), the standard deviation of the \( z \)-scores is \( 1 \), and the \( z \)-scores are numbers without units of measurement.

\(\boxed{\text{Mean} = 0, \text{Standard Deviation} = 1, \text{Units} = \text{none}}\)

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