Alright, let's break down the process of calculating the z-score step-by-step.
1. Understand the given data:
- The mean [tex]\( \mu \)[/tex] is 7.
- The variance [tex]\( \sigma^2 \)[/tex] is 12.405, which means the standard deviation [tex]\( \sigma \)[/tex] is approximately 3.522.
- You are given a data point [tex]\( x \)[/tex] which is 11.7.
2. Identify the formula for the z-score:
The z-score formula is:
[tex]\[
z_x = \frac{x - \mu}{\sigma}
\][/tex]
where [tex]\( x \)[/tex] is the data point, [tex]\( \mu \)[/tex] is the mean, and [tex]\( \sigma \)[/tex] is the standard deviation.
3. Substitute the values into the formula:
First, identify each component:
[tex]\[
x = 11.7
\][/tex]
[tex]\[
\mu = 7
\][/tex]
[tex]\[
\sigma = 3.522
\][/tex]
4. Perform the calculations:
Substitute [tex]\( x \)[/tex], [tex]\( \mu \)[/tex], and [tex]\( \sigma \)[/tex] into the z-score formula:
[tex]\[
z_x = \frac{11.7 - 7}{3.522}
\][/tex]
Calculate the numerator:
[tex]\[
11.7 - 7 = 4.7
\][/tex]
Now, divide the numerator by the standard deviation:
[tex]\[
z_x = \frac{4.7}{3.522} \approx 1.334
\][/tex]
So, the z-score for the data point 11.7 is approximately 1.334.
