Question

a manufacturer must test that his bolts are 4.00 cm long when they come off the assembly line. he must recalibrate his machines if the bolts are too long or too short. after sampling 169 randomly - selected bolts off the assembly line, he calculates the sample mean to be 4.07 cm. he knows that the population standard deviation is 0.45 cm. assuming a level of significance of 0.02, is there sufficient evidence to show that the manufacturer needs to recalibrate the machines? step 2 of 3: compute the value of the test statistic. round your answer to two decimal places.

Explanation:

Step1: Identify the formula

The formula for the z - test statistic in a one - sample z - test is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x}$ is the sample mean, $\mu$ is the population mean, $\sigma$ is the population standard deviation, and $n$ is the sample size.

Step2: Substitute the values

We are given that $\bar{x} = 4.07$, $\mu=4.00$, $\sigma = 0.45$, and $n = 169$. First, calculate $\sqrt{n}=\sqrt{169}=13$. Then, $\frac{\sigma}{\sqrt{n}}=\frac{0.45}{13}\approx0.0346$. Now, $z=\frac{4.07 - 4.00}{0.0346}=\frac{0.07}{0.0346}\approx2.02$.

Answer:

$2.02$