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 3 of 3: draw a conclusion and interpret the decision. answer we reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance that the mean length of the bolts is different from 4.00 cm and the machines need to be recalibrated. we fail to reject the null hypothesis and conclude that there is sufficient evidence at a 0.02 level of significance that the mean length of the bolts is different from 4.00 cm and the machines need to be recalibrated. we reject the null hypothesis and conclude that there is sufficient evidence at a 0.02 level of significance that the mean length of the bolts is different from 4.00 cm and the machines need to be recalibrated. we fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance that the mean length of the bolts is different from 4.00 cm and the machines need to be recalibrated.

Explanation:

Step1: Set up hypotheses

The null hypothesis $H_0:\mu = 4.00$ and the alternative hypothesis $H_1:\mu
eq4.00$.

Step2: Calculate the test - statistic

The formula for the z - test statistic in a one - sample z - test is $z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, where $\bar{x} = 4.07$, $\mu = 4.00$, $\sigma=0.45$, and $n = 169$. So $z=\frac{4.07 - 4.00}{\frac{0.45}{\sqrt{169}}}=\frac{0.07}{\frac{0.45}{13}}=\frac{0.07\times13}{0.45}=\frac{0.91}{0.45}\approx2.02$.
The critical values for a two - tailed test with $\alpha = 0.02$ are $z_{\alpha/2}=\pm 2.33$. Since $|2.02|<2.33$, we fail to reject the null hypothesis.

Step3: Draw conclusion

Failing to reject the null hypothesis means there is insufficient evidence at the 0.02 level of significance that the mean length of the bolts is different from 4.00 cm, so the machines do not need to be recalibrated.

Answer:

We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance that the mean length of the bolts is different from 4.00 cm and the machines do not need to be recalibrated. (The closest option among the given ones is: We fail to reject the null hypothesis and conclude that there is insufficient evidence at a 0.02 level of significance that the mean length of the bolts is different from 4.00 cm and the machines need to be recalibrated, although the last part about the machines needing to be recalibrated is incorrect in our conclusion but it's the closest in terms of the hypothesis - testing part)