Question

z - scores, proportions, and percentiles

1. a normal distribution of scores has a standard deviation of 10. find the z - scores corresponding to each of the following values:

a) a score that is 20 points above the mean.
b) a score that is 10 points below the mean.
c) a score that is 15 points above the mean
d) a score that is 30 points below the mean.

Explanation:

Part (a)

Step1: Recall z - score formula

The formula for a z - score is $z=\frac{X - \mu}{\sigma}$, where $X$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation. If a score is 20 points above the mean, then $X-\mu = 20$ and $\sigma = 10$.

Step2: Substitute values into formula

Substitute $X - \mu=20$ and $\sigma = 10$ into the z - score formula: $z=\frac{20}{10}=2$

Part (b)

Step1: Determine $X-\mu$

If a score is 10 points below the mean, then $X-\mu=- 10$ (negative because it is below the mean) and $\sigma = 10$.

Step2: Calculate z - score

Using the z - score formula $z=\frac{X - \mu}{\sigma}$, substitute $X - \mu=-10$ and $\sigma = 10$: $z=\frac{- 10}{10}=-1$

Part (c)

Step1: Identify $X-\mu$

A score that is 15 points above the mean means $X - \mu = 15$ and $\sigma=10$.

Step2: Compute z - score

Substitute into the z - score formula: $z=\frac{15}{10}=1.5$

Part (d)

Answer:

s:
a) $\boldsymbol{2}$
b) $\boldsymbol{-1}$
c) $\boldsymbol{1.5}$
d) $\boldsymbol{-3}$