Question

1. Two machines are used for filling plastic bottles (for a desired volume of 16 ounces). The fill volume can be assumed Normal, with standard deviations σ1=0.020 and σ2=0.025 ounces for the two machines respectively. The quality control engineer wants to check if both machines fill the bottles to the same volume or if the volumes differ. A random sample of 10 bottles is taken from each machine. Assume the data are independent. The data are given below:

Machine_01			Machine_02
16.03	16.01		16.02	16.03
16.04	15.96		15.97	16.04
16.05	15.98		15.96	16.02
16.05	16.02		16.01	16.01
16.02	15.99		15.99	16.00

2. What is the type of statistical test procedure that should be used to test the hypotheses? Explain.

3. Construct a 95% confidence interval for the true mean difference. Test the hypotheses using the confidence interval. Interpret the test result clearly.

4. Test the hypotheses using the Test Statistic / Critical Value method (you must clearly indicate the test statistic, and the critical value(s) use Take α=5%). Do you get the same answer as part (c)? If your answer is the same as (c), you do not have to interpret again. Otherwise, interpret the test findings.

5. What is the P value of the test? Test the hypotheses using the P value. Take α=5%. Do you get the same answer as parts (c) and (d)? If your answer is the same as (c) and (d), you do not have to interpret again. Otherwise, interpret the test findings.

Answer 1

b. What is the type of statistical test procedure that should be used to test the hypotheses? Explain.

Here, the bottles are drawn from a population that follow a Normal distribution and the variances are known. Hence, to test teh hypothesis, we shall use the Z test for testing.

We shall be testing the :

  

We shall calculate the mean

	Machine1	Machine2
	16.03	16.02
	16.04	15.97
	16.05	15.96
	16.05	16.01
	16.02	15.99
	16.01	16.03
	15.96	16.04
	15.98	16.02
	16.02	16.01
	15.99	16
Mean	16.015	16.005

We are given the population SD :

c. The 95% confidence interval for difference in means is given by:

The confidence interval contains the value 0. Hence, we fail to reject the null hypothesis and we conclude that the two machines  fill the bottles to the same volume.

c). The test statistic is given by  

  

The critical value of Z at 5% level is 1.96.

Since the calculated value of Z>the critical value, we fail to reject the null hypothesis.

d). The calculated Z value is <the critical value, we fail to reject the Null hypothesis. Yes, I am getting the same answer that of question c.

e). The p-value of the test is 0.3233 (=2*(1-NORM.S.DIST(0.9877,TRUE)) function in EXCEL). Since the p-value >0.05, we fail to reject the null hypothesis.