Question

Given that the mean of a normal variate $X$ is 9 and standard deviation is 3, then find:
(i) the z-score of the data point 15
(ii) the data point if its z-score is 4

Hint:

Use $z = \dfrac{x - \mu}{\sigma}$ to convert data into standard scores. Rearrange it as $x = \mu + z\sigma$ to find the raw score from a given $z$-score.

Answer 1

We use the formula for computing a $z$-score:
\[ z = \frac{x - \mu}{\sigma} \]
Where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation.
Given: $\mu = 9$, $\sigma = 3$
Part (i): Given $x = 15$
\[ z = \frac{15 - 9}{3} = \frac{6}{3} = 2 \]
So, the $z$-score is $\boxed{2}$.
Part (ii): We are given $z = 4$ and want to find $x$. Use the inverse formula:
\[ x = \mu + z.\sigma = 9 + 4.3 = 9 + 12 = \boxed{21} \]