Question

consider a sample with a mean equal to 47 and a standard deviation equal to 11. calculate the z - scores for the following values.
a) 59
b) 76
c) 42
d) 15

Explanation:

The formula for calculating the z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is the value from the dataset, $\mu$ is the mean, and $\sigma$ is the standard deviation. We are given that $\mu = 47$ and $\sigma=11$. We will calculate the z - score for each of the given $x$ values (59, 76, 42, 15) one by one.

Step 1: Calculate the z - score for $x = 59$

We substitute $x = 59$, $\mu=47$ and $\sigma = 11$ into the z - score formula.
$z=\frac{59 - 47}{11}=\frac{12}{11}\approx1.09$

Step 2: Calculate the z - score for $x = 76$

Substitute $x = 76$, $\mu = 47$ and $\sigma=11$ into the formula.
$z=\frac{76 - 47}{11}=\frac{29}{11}\approx2.64$

Step 3: Calculate the z - score for $x = 42$

Substitute $x = 42$, $\mu = 47$ and $\sigma = 11$ into the formula.
$z=\frac{42-47}{11}=\frac{- 5}{11}\approx - 0.45$

Step 4: Calculate the z - score for $x=15$

Substitute $x = 15$, $\mu=47$ and $\sigma = 11$ into the formula.
$z=\frac{15 - 47}{11}=\frac{-32}{11}\approx - 2.91$

Answer:

• For $x = 59$: $z\approx1.09$
• For $x = 76$: $z\approx2.64$
• For $x = 42$: $z\approx - 0.45$
• For $x = 15$: $z\approx - 2.91$