Question

A CNC lathe is programmed to cut a groove that is 18.5mmin depth. A sample of 12 parts was collected to evaluate whether the lathe is producing the correct depth. From the sample, the mean depth was 19.8 mm and it is known that the standard deviation of the population, σ,is 1.5 mm.

a)Perform a hypothesis test to determine whether there is evidence from the sample of 12 parts that the depth of the grooves being produced is not18.5 mm. Assume a value of 0.05 for α.For your hypothesis tests, show your work for all 7 steps of the hypothesis test and label the steps (1, 2, ...). For step 7, state your conclusion using all 3 items specified for problem 5.
b)Determine the p-value for this test.
c)Determine the power of the test to determine that there has been a shift to μ new= 20 mm.

Answer 1

To determine whether there is evidence that the depth of the grooves produced by the lathe is not 18.5 mm, we can perform a hypothesis test. We state the hypotheses, set the significance level, select the test statistic, formulate the decision rule, calculate the test statistic, make a decision, and state the conclusion.

Step-by-step explanation:

Step 1: State the hypotheses

The null hypothesis, denoted as H0, is that the depth of the grooves produced by the lathe is 18.5 mm. The alternative hypothesis, denoted as Ha, is that the depth of the grooves is not 18.5 mm.

Step 2: Set the significance level

The significance level, also known as alpha (α), is given as 0.05.

Step 3: Select the test statistic

Since we are comparing the mean depth of the grooves, we will use a t-test. The test statistic for a t-test is given by:

t = (sample mean - hypothesized mean) / (sample standard deviation / √n)

Step 4: Formulate the decision rule

Using the significance level of 0.05 and the degrees of freedom (n-1 = 12-1 = 11), we can find the critical value for a two-tailed test from the t-distribution table. The critical value will tell us the range of values that would lead us to reject the null hypothesis.

Step 5: Calculate the test statistic

Using the given sample mean (19.8 mm) and the known standard deviation of the population (1.5 mm), we can substitute these values into the formula for the t-statistic to calculate its value.

Step 6: Make a decision

Compare the calculated test statistic with the critical value. If the calculated test statistic falls outside the range provided by the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Step 7: State the conclusion

In this case, we would state the conclusion as follows: There is (not) enough evidence from the sample of 12 parts to support the claim that the depth of the grooves being produced by the lathe is not 18.5 mm.