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

The calculation of a $z$-score utilizes the formula: \[z = \frac{x - \mu}{\sigma}\] Here, $x$ represents the data point, $\mu$ denotes the mean, and $\sigma$ signifies the standard deviation.
Provided values: $\mu = 9$, $\sigma = 3$
Part (i): With $x = 15$ provided, the calculation is: \[z = \frac{15 - 9}{3} = \frac{6}{3} = 2\]
Consequently, the $z$-score is $\boxed{2}$.
Part (ii): Given $z = 4$, we seek the corresponding $x$ value by applying the inverse formula: \[x = \mu + z.\sigma = 9 + 4.3 = 9 + 12 = \boxed{21}\]