Question

all scenarios are normally distributed. 1. what is the z - score of a value of 40 with a mean of 50 and a standard deviation of 5? 3. the prices of a set of prom dresses have a mean of $108 and a standard deviation of 18. if a dress in this set has a z - score of 1.5, find its price.

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Substitute values for first question

Given $x = 40$, $\mu=50$, and $\sigma = 5$. Substitute into the formula: $z=\frac{40 - 50}{5}$.

Step3: Calculate the z - score for first question

$z=\frac{- 10}{5}=-2$.

Step4: Rearrange z - score formula for second question

From $z=\frac{x-\mu}{\sigma}$, we can solve for $x$: $x=z\sigma+\mu$.

Step5: Substitute values for second question

Given $z = 1.5$, $\mu = 108$, and $\sigma=18$. Substitute into the formula: $x=1.5\times18 + 108$.

Step6: Calculate the value for second question

$x = 27+108=135$.

Answer:

1. The z - score is - 2.
2. The price of the dress is $\$135$.