To answer this question, we need to understand the concept of converting a data set to [tex]$z$[/tex] scores. This conversion allows us to standardize the data, which simplifies comparison between different sets of data.
### Conversion to [tex]$z$[/tex] Scores:
The formula to convert a data value [tex]\( x \)[/tex] to a [tex]$z$[/tex] score is given by:
[tex]\[ z = \frac{(x - \mu)}{\sigma} \][/tex]
where:
- [tex]\( x \)[/tex] is the original data value,
- [tex]\( \mu \)[/tex] is the mean of the original data set,
- [tex]\( \sigma \)[/tex] is the standard deviation of the original data set.
### Step-by-Step Solution:
1. Mean Calculation:
When converting an entire data set to [tex]$z$[/tex] scores, each value [tex]$x$[/tex] in our data set is standardized by subtracting the mean [tex]\(\mu\)[/tex] and then dividing by the standard deviation [tex]\(\sigma\)[/tex]. Because we adjust each data point in the same manner, the new mean of our [tex]$z$[/tex] score distribution will always be 0.
[tex]\[
\mu_{z} = 0
\][/tex]
2. Standard Deviation Calculation:
Standardizing does not change the spread of the distribution in a relative sense; it only adjusts the scale. Hence, the standard deviation of a distribution of [tex]$z$[/tex] scores is always 1, as we convert every standard deviation unit of the original distribution into exactly one unit on the [tex]$z$[/tex] score scale.
[tex]\[
\sigma_{z} = 1
\][/tex]
### Conclusion:
When all pulse rates of women are converted to [tex]$z$[/tex] scores:
- The mean of the [tex]$z$[/tex] scores ([tex]\(\mu_{z}\)[/tex]) becomes [tex]\(\boxed{0}\)[/tex].
- The standard deviation of the [tex]$z$[/tex] scores ([tex]\(\sigma_{z}\)[/tex]) becomes [tex]\(\boxed{1}\)[/tex].
