Question

on a mid - term exam the average of the scores was 70 and the sd was 10.
a. convert each of the following scores to standard units: i. 82 ii. 44 iii. 95
b. find the test scores for which the standard units are: i. 0 ii. 1.3 iii. - 2.1

Explanation:

Step1: Recall z - score formula

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is the raw score, $\mu$ is the mean, and $\sigma$ is the standard deviation. Given $\mu = 70$ and $\sigma=10$.

Step2: Calculate z - score for $x = 82$

Substitute $x = 82$, $\mu = 70$, and $\sigma = 10$ into the formula: $z=\frac{82 - 70}{10}=\frac{12}{10}=1.2$.

Step3: Calculate z - score for $x = 44$

Substitute $x = 44$, $\mu = 70$, and $\sigma = 10$ into the formula: $z=\frac{44 - 70}{10}=\frac{- 26}{10}=-2.6$.

Step4: Calculate z - score for $x = 95$

Substitute $x = 95$, $\mu = 70$, and $\sigma = 10$ into the formula: $z=\frac{95 - 70}{10}=\frac{25}{10}=2.5$.

Step5: Find raw score for $z = 0$

Using the formula $z=\frac{x - \mu}{\sigma}$, when $z = 0$, we have $0=\frac{x - 70}{10}$. Cross - multiply gives $0=x - 70$, so $x = 70$.

Step6: Find raw score for $z = 1.3$

Using the formula $z=\frac{x - \mu}{\sigma}$, substitute $z = 1.3$, $\mu = 70$, and $\sigma = 10$. Then $1.3=\frac{x - 70}{10}$. Cross - multiply: $1.3\times10=x - 70$. So $x=13 + 70=83$.

Step7: Find raw score for $z=-2.1$

Using the formula $z=\frac{x - \mu}{\sigma}$, substitute $z=-2.1$, $\mu = 70$, and $\sigma = 10$. Then $-2.1=\frac{x - 70}{10}$. Cross - multiply: $-2.1\times10=x - 70$. So $x=-21 + 70 = 49$.

Answer:

a. $z = 1.2$ for $x = 82$, $z=-2.6$ for $x = 44$, $z = 2.5$ for $x = 95$
b. $x = 70$ for $z = 0$, $x = 83$ for $z = 1.3$, $x = 49$ for $z=-2.1$