To find the height of a man given the [tex]$z$[/tex]-score of -0.46, we will use the formula that relates [tex]$z$[/tex]-scores to the individual data points in a normally distributed dataset:
[tex]\[ \text{height} = \text{mean height} + (z \times \text{standard deviation}) \][/tex]
Here are the given values:
- Mean height (\(\mu\)): 69.0 inches
- Standard deviation (\(\sigma\)): 2.8 inches
- \(z\)-score (\(z\)): -0.46
Now, we substitute the values into the formula:
[tex]\[
\text{height} = 69.0 + (-0.46 \times 2.8)
\][/tex]
Calculate the product of the \(z\)-score and the standard deviation:
[tex]\[
-0.46 \times 2.8 = -1.288
\][/tex]
Next, add this result to the mean height:
[tex]\[
69.0 + (-1.288) = 67.712
\][/tex]
Finally, we round this result to one decimal place:
[tex]\[
67.712 \approx 67.7
\][/tex]
Therefore, the height of the man with a [tex]$z$[/tex]-score of -0.46 is:
[tex]\[
\boxed{67.7} \text{ inches}
\][/tex]
