Question

a data set has the following characteristics:
mean: 4.9
median: 6
mode: 6
variance: 4
using the formula, calculate the z - score for the listed data points.
z - score: $z_x=\frac{x - mu}{sigma}$
$z_1=square$
$z_6 = 0.05$
$z_{6.5}=square$

Explanation:

Step1: Find the standard deviation (σ)

We know that variance \( \text{Var} = \sigma^2 \). Given variance is 4, so \( \sigma = \sqrt{4} = 2 \). The mean \( \mu = 4.9 \).

Step2: Calculate \( z_1 \)

For \( x = 1 \), use the z - score formula \( z_x=\frac{x - \mu}{\sigma} \).
Substitute \( x = 1 \), \( \mu = 4.9 \), \( \sigma = 2 \) into the formula:
\( z_1=\frac{1 - 4.9}{2}=\frac{- 3.9}{2}=-1.95 \)

Step3: Verify \( z_5 \) (optional, but we can check)

For \( x = 5 \), \( z_5=\frac{5 - 4.9}{2}=\frac{0.1}{2}=0.05 \), which matches the given value.

Step4: Calculate \( z_{6.5} \)

For \( x = 6.5 \), use the z - score formula \( z_x=\frac{x - \mu}{\sigma} \).
Substitute \( x = 6.5 \), \( \mu = 4.9 \), \( \sigma = 2 \) into the formula:
\( z_{6.5}=\frac{6.5 - 4.9}{2}=\frac{1.6}{2}=0.8 \)

Answer:

\( z_1=-1.95 \), \( z_{6.5}=0.8 \) (and \( z_5 = 0.05 \) as given)