Answer:
z = -0.4.
Step-by-step explanation:
Here's the formula for finding the z-score (a.k.a. standardized normal variable) of one measurement [tex]x[/tex] of a normal random variable [tex]X \sim N(\mu,\; \sigma^{2})[/tex]:
[tex]\displaystyle z = \frac{x-\mu}{\sigma}[/tex],
where
- [tex]z[/tex] is the z-score of this measurement,
- [tex]x[/tex] is the value of this measurement,
- [tex]\mu[/tex] is the mean of the random normal variable, and
- [tex]\sigma[/tex] is the standard deviation (the square root of variance) of the random normal variable.
Let the length of a trout in this lake be [tex]X[/tex] inches.
[tex]X \sim N(30, \; 4.5^{2})[/tex].
- [tex]\mu = 30[/tex], and
- [tex]\sigma = 4.5[/tex].
For this measurement, [tex]x = 28.2[/tex]. Apply the formula to get the z-score:
[tex]\displaystyle z = \frac{x - \mu}{\sigma} = \frac{28.2 - 30}{4.5} = -0.4[/tex].
