Question

an algebra 2 teacher wants her students to practice using z - scores, so she told her class their test scores as z - scores. find each person’s score, given their z - score if the class mean is 84 points and the standard deviation is 3.8 points. gretl: ( z = -2.37 ) mia: ( z = 0.79 ) lucca: ( z = -3.68 ) isaac: ( z = 2.89 ) 70 75 87 95 79 68 37 89

Explanation:

The formula to convert a z - score to a raw score ($x$) is $x=\mu + z\times\sigma$, where $\mu$ is the mean, $z$ is the z - score, and $\sigma$ is the standard deviation. Here, $\mu = 84$ and $\sigma=3.8$.

Step 1: Find Gretl's score

We know that for Gretl, $z=- 2.37$, $\mu = 84$ and $\sigma = 3.8$.
Using the formula $x=\mu+z\times\sigma$, we substitute the values:
$x = 84+(-2.37)\times3.8$
First, calculate $(-2.37)\times3.8=-8.996$
Then, $x=84 - 8.996 = 75.004\approx75$

Step 2: Find Mia's score

For Mia, $z = 0.79$, $\mu=84$ and $\sigma = 3.8$.
Using the formula $x=\mu + z\times\sigma$:
$x=84+0.79\times3.8$
Calculate $0.79\times3.8 = 3.002$
Then, $x=84 + 3.002=87.002\approx87$

Step 3: Find Lucca's score

For Lucca, $z=-3.68$, $\mu = 84$ and $\sigma=3.8$.
Using the formula $x=\mu+z\times\sigma$:
$x = 84+(-3.68)\times3.8$
First, calculate $(-3.68)\times3.8=-14.0$ (approx, more accurately $- 14.0$ (since $3.68\times3.8 = 14.0$))
Then, $x=84-14.0 = 70.0$? Wait, let's calculate precisely: $3.68\times3.8=(3 + 0.68)\times3.8=3\times3.8+0.68\times3.8 = 11.4+2.584 = 13.984$
So $x = 84-13.984=70.016\approx70$? Wait, but let's check again. Wait, maybe I made a mistake. Wait, $z=-3.68$, $\sigma = 3.8$, $\mu = 84$.
$x=84+(-3.68)\times3.8=84 - 3.68\times3.8$
$3.68\times3.8$: $3\times3.8 = 11.4$, $0.68\times3.8=2.584$, so $3.68\times3.8 = 13.984$
$84-13.984 = 70.016\approx70$? But the options have 68,70 etc. Wait, maybe my calculation is wrong. Wait, $z=-3.68$, $\sigma = 3.8$, $\mu = 84$.
$x=84+z\sigma=84+(-3.68)\times3.8=84 - 13.984 = 70.016\approx70$

Wait, but let's check Lucca's score again. Wait, maybe the standard deviation is 3.8, mean 84. Let's recalculate:

Wait, $z=-3.68$, so $x = 84+(-3.68)\times3.8=84 - 13.984 = 70.016\approx70$

Step 4: Find Isaac's score

For Isaac, $z = 2.89$, $\mu=84$ and $\sigma = 3.8$.
Using the formula $x=\mu+z\times\sigma$:
$x=84 + 2.89\times3.8$
Calculate $2.89\times3.8=(2 + 0.89)\times3.8=2\times3.8+0.89\times3.8=7.6 + 3.382=10.982$
Then, $x=84 + 10.982=94.982\approx95$

Answer:

• Gretl's score: 75 (since $x = 84+(-2.37)\times3.8\approx75$)
• Mia's score: 87 (since $x = 84 + 0.79\times3.8\approx87$)
• Lucca's score: 70 (since $x=84+(-3.68)\times3.8\approx70$)
• Isaac's score: 95 (since $x = 84+2.89\times3.8\approx95$)