To find the value of the test statistic [tex]\( z \)[/tex] using the provided formula, let's go through the steps one by one.
### Step 1: Identify the given values
- Sample mean ([tex]\( \bar{x} \)[/tex]): [tex]\( 8.91 \, \text{cm} \)[/tex]
- Population mean ([tex]\( \mu \)[/tex]): [tex]\( 9 \, \text{cm} \)[/tex]
- Population standard deviation ([tex]\( \sigma \)[/tex]): [tex]\( 0.36 \, \text{cm} \)[/tex]
- Sample size ([tex]\( n \)[/tex]): [tex]\( 72 \)[/tex]
### Step 2: Calculate the standard error of the mean
The standard error of the mean (SEM) is calculated using the formula:
[tex]\[
\text{SEM} = \frac{\sigma}{\sqrt{n}}
\][/tex]
Plugging in the given values:
[tex]\[
\text{SEM} = \frac{0.36}{\sqrt{72}}
\][/tex]
[tex]\[
\sqrt{72} \approx 8.485
\][/tex]
[tex]\[
\text{SEM} = \frac{0.36}{8.485} \approx 0.042426
\][/tex]
### Step 3: Calculate the test statistic [tex]\( z \)[/tex]
The test statistic [tex]\( z \)[/tex] is calculated using the formula:
[tex]\[
z = \frac{\bar{x} - \mu}{\text{SEM}}
\][/tex]
Replacing the variables with the given values:
[tex]\[
z = \frac{8.91 - 9}{0.042426}
\][/tex]
[tex]\[
z = \frac{-0.09}{0.042426} \approx -2.12
\][/tex]
### Step 4: Select the correct option
The calculated test statistic [tex]\( z \)[/tex] is approximately [tex]\( -2.12 \)[/tex]. Therefore, the correct answer from the given options is:
[tex]\[
\boxed{-2.12}
\][/tex]
