Question

compare the z - scores to decide which of the following x - values lie the greatest distance above the mean and the greatest distance below the mean
a. x = 100, μ = 50, σ = 25
b. x = 1, μ = 4, σ = 1
c. x = 0, μ = 200, σ = 100
d. x = 10, μ = 5, σ = 3
a. the z - score is 2.00. (type an integer or decimal rounded to two decimal places as needed.)
b. the z - score is - 3.00. (type an integer or decimal rounded to two decimal places as needed.)
c. the z - score is. (type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall z - score formula

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

Step2: Calculate z - score for part c

Given $x = 0$, $\mu=200$, and $\sigma = 100$. Substitute these values into the z - score formula:
$z=\frac{0 - 200}{100}=\frac{-200}{100}=-2.00$

Step3: Compare z - scores

We have:

• For part a: $z = 2.00$
• For part b: $z=-3.00$
• For part c: $z=-2.00$
• For part d: First, calculate the z - score. Given $x = 10$, $\mu = 5$, $\sigma=3$. Then $z=\frac{10 - 5}{3}=\frac{5}{3}\approx1.67$

The greatest distance above the mean is indicated by the largest positive z - score. Here, the largest positive z - score is $z = 2.00$ (from part a). The greatest distance below the mean is indicated by the smallest (most negative) z - score. Here, the smallest z - score is $z=-3.00$ (from part b).

Answer:

The x - value that lies the greatest distance above the mean is $x = 100$ (from part a) and the x - value that lies the greatest distance below the mean is $x = 1$ (from part b).