A watch designer claims that men have wrist breadths with a mean equal to 9 cm. A simple random sample of wrist breadths of 72 men has a mean of 8.91 cm. The population standard deviation is 0.36 cm. Assume a confidence level of [tex]$a=0.01$[/tex].

Find the value of the test statistic [tex]$z$[/tex] using:

[tex]
z=\frac{\bar{x}-\mu_{\bar{x}}}{\frac{\sigma}{\sqrt{n}}}
[/tex]

A. [tex][tex]$-1.27$[/tex][/tex]
B. [tex]0.06[/tex]
C. [tex]$-2.12$[/tex]
D. [tex]2.12[/tex]



Answer :

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]