Calculate a z-score using the rainforest data:

Mean (μ) = 7
Variance (σ²) = 12.405
Standard Deviation (σ) ≈ 3.522

Using the formula [tex]\( z = \frac{x - \mu}{\sigma} \)[/tex], find the z-score for the data point 11.7:

[tex]\( x = \_\_\_ \)[/tex]
[tex]\( \mu = \_\_\_ \)[/tex]



Answer :

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.