Question

A manufacturer must test that his bolts are 2.00cm long when they come off the assembly line. He must recalibrate his machines if the bolts are too long or too short. After sampling 144 randomly selected bolts off the assembly line, he calculates the sample mean to be 2.11cm. He knows that the population standard deviation is 0.56cm. Assuming a level of significance of 0.05, is there sufficient evidence to show that the manufacturer needs to recalibrate the machines? State the null and alternative hypotheses for the test. Fill in the blank below. H0Ha: μ=2.00: μ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯2.00

Answer 1

To determine if the manufacturer needs to recalibrate the machines, we can conduct a hypothesis test. The null hypothesis is that the mean length of the bolts is 2.00cm, while the alternative hypothesis is that the mean length of the bolts is not equal to 2.00cm. By calculating the test statistic and comparing it to the critical value, we can make a conclusion regarding the need for recalibration.

Step-by-step explanation:

To determine if the manufacturer needs to recalibrate the machines, we can conduct a hypothesis test.

The null hypothesis, denoted as H0, is that the mean length of the bolts is 2.00cm, or μ = 2.00cm. The alternative hypothesis, denoted as Ha, is that the mean length of the bolts is not equal to 2.00cm, or μ ≠ 2.00cm.

Next, we can calculate the test statistic, which is the z-score when the population standard deviation is known. The formula for the test statistic is z = (sample mean - population mean) / (population standard deviation / √sample size). In this case, the sample mean is 2.11cm, the population mean is 2.00cm, the population standard deviation is 0.56cm, and the sample size is 144. Plugging these values into the formula, we can calculate the test statistic.

Finally, we can compare the test statistic to the critical value based on the level of significance, which is 0.05 in this case. If the test statistic falls outside of the critical value range, we reject the null hypothesis and conclude that there is sufficient evidence to show that the manufacturer needs to recalibrate the machines.