Sure, let's go through the solutions step by step.
### Part (a): Calculating the z-score for a height of 61 inches
To find the z-score for a height [tex]\( X \)[/tex] when given the mean height [tex]\( \mu \)[/tex] and the standard deviation [tex]\( \sigma \)[/tex], we use the formula:
[tex]\[ z = \frac{X - \mu}{\sigma} \][/tex]
Here,
- [tex]\( X \)[/tex] (the height in question) is 61 inches.
- [tex]\( \mu \)[/tex] (the mean height) is 64 inches.
- [tex]\( \sigma \)[/tex] (the standard deviation) is 2.7 inches.
Substitute these values into the formula:
[tex]\[ z = \frac{61 - 64}{2.7} \][/tex]
[tex]\[ z = \frac{-3}{2.7} \][/tex]
[tex]\[ z \approx -1.111 \][/tex]
So, the z-score for a height of 61 inches is approximately -1.111.
### Part (b): Determining the height corresponding to a z-score of 0
The z-score of 0 indicates that the value is exactly at the mean.
Given:
- Mean height [tex]\( \mu \)[/tex] is 64 inches.
Since the z-score of 0 corresponds to the mean height, a woman with a z-score of 0 would have a height equal to the mean height, which is 64 inches.
### Summary
a. The z-score for a height of 61 inches is approximately -1.111.
b. The height of a woman with a z-score of 0 is 64 inches.
