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
(type an integer or decimal rounded to two decimal places as needed.)

Explanation:

Step1: Recall z - score formula

The formula for the z - score 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 a

For $x = 100$, $\mu=50$, and $\sigma = 25$, we substitute into the formula: $z=\frac{100 - 50}{25}=\frac{50}{25}=2.00$.

Step3: Calculate z - score for part b

For $x = 1$, $\mu = 4$, and $\sigma=1$, we have $z=\frac{1 - 4}{1}=\frac{- 3}{1}=-3.00$.

Step4: Calculate z - score for part c

For $x = 0$, $\mu = 200$, and $\sigma = 100$, we get $z=\frac{0 - 200}{100}=\frac{-200}{100}=-2.00$.

Step5: Calculate z - score for part d

For $x = 10$, $\mu = 5$, and $\sigma = 3$, we obtain $z=\frac{10 - 5}{3}=\frac{5}{3}\approx1.67$.

The greatest distance above the mean is represented by the largest positive z - score, and the greatest distance below the mean is represented by the smallest (most negative) z - score.

Answer:

The value with the greatest distance above the mean is $x = 100$ (with $z = 2.00$) and the value with the greatest distance below the mean is $x = 1$ (with $z=-3.00$).