To calculate the z-scores for the heights provided, we can use the z-score formula:
\[ z = \frac{(X - \mu)}{\sigma} \]
where:
- \(X\) is the value for which we are calculating the z-score,
- \(\mu\) is the mean,
- \(\sigma\) is the standard deviation.
a) For the man who is 6 feet 3 inches tall
First, convert 6 feet 3 inches to inches:
\[ 6 \text{ feet} = 6 \times 12 = 72 \text{ inches} \] \[ 6 \text{ feet 3 inches} = 72 + 3 = 75 \text{ inches} \]
Now, using the mean (\(\mu\) = 69.7 inches) and the standard deviation (\(\sigma\) = 2.65 inches):
\[ z = \frac{(75 - 69.7)}{2.65} = \frac{5.3}{2.65} \approx 2.00 \]
So, the man's z-score is:
\[ z = 2.00 \]
b) For the woman who is 5 feet 11 inches tall
Convert 5 feet 11 inches to inches:
\[ 5 \text{ feet} = 5 \times 12 = 60 \text{ inches} \] \[ 5 \text{ feet 11 inches} = 60 + 11 = 71 \text{ inches} \]
Using the mean (\(\mu\) = 64.7 inches) and the standard deviation (\(\sigma\) = 2.54 inches):
\[ z = \frac{(71 - 64.7)}{2.54} = \frac{6.3}{2.54} \approx 2.48 \]
So, the woman's z-score is:
\[ z = 2.48 \]
c) Who is relatively taller?
To determine who is relatively taller, we compare the z-scores:
- The man's z-score is 2.00
- The woman's z-score is 2.48
The woman has a higher z-score, indicating that she is relatively taller when compared to her gender's height distribution.
Thus, the answer is:
The 5 foot 11 inch American woman.