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. greti: ( z = -2.37 ) mia: ( z = 0.79 ) lucca: ( z = -3.68 ) isaac: ( z = 2.80 ) options: 70, 75, 87, 95, 79, 68, 37, 89

Explanation:

To find each person's score, we use the formula for converting a z - score to a raw score: \( x=\mu + z\sigma \), where \( \mu = 84 \) (the mean), \( \sigma=3.8 \) (the standard deviation), and \( z \) is the z - score.

Step 1: Calculate Greti's score

We know that for Greti, \( z=- 2.37 \), \( \mu = 84 \), and \( \sigma = 3.8 \).
Substitute these values into the formula \( x=\mu+z\sigma \):
\( x = 84+(-2.37)\times3.8=84 - 2.37\times3.8 \)
First, calculate \( 2.37\times3.8 = 9.006 \)
Then, \( x=84 - 9.006=74.994\approx75 \)

Step 2: Calculate Mia's score

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

Step 3: Calculate Lucca's score

For Lucca, \( z=-3.68 \), \( \mu = 84 \), \( \sigma=3.8 \)
Using the formula \( x=\mu+z\sigma \):
\( x = 84+(-3.68)\times3.8=84-3.68\times3.8 \)
Calculate \( 3.68\times3.8=14.0\) (approx, more accurately \( 3.68\times3.8 = 14.0\) (exact value: \(3.68\times3.8=(3 + 0.68)\times3.8=3\times3.8+0.68\times3.8 = 11.4+2.584 = 13.984\)))
\( x=84 - 13.984 = 70.016\approx70 \)

Step 4: Calculate Isaac's score

For Isaac, \( z = 2.80 \), \( \mu=84 \), \( \sigma=3.8 \)
Using the formula \( x=\mu+z\sigma \):
\( x=84+2.80\times3.8 \)
Calculate \( 2.80\times3.8 = 10.64 \)
Then, \( x=84 + 10.64=94.64\approx95 \)

Answer:

Greti: 75, Mia: 87, Lucca: 70, Isaac: 95