Department of Engineering Management, Information and Systems EMIS 7370 STAT 5340 Department of Engineering Management, Information and Systems Probability and Statistics for Scientists and Engineers Tests of Hypothesis Tests of Means and Variances Dr. Jerrell T. Stracener

Example A company produces and markets coffee in cans which are advertised as containing one pound of coffee. What this means is that the true mean weight of coffee per can is 1 pound. If the true mean weight of coffee per can exceeds 1 pound, the company’s profit will suffer. On the other hand, if the true mean weight is very much less than 1 pound, consumers will complain and sales may decrease. To monitor the process, 25 cans of coffee are randomly selected during each day’s production. The process will be adjusted if there is evidence to indicate that the true mean amount of coffee is not 1 pound.

Example A decision rule is desired so that the probability of adjusting the process when the true mean weight of coffee per can is equal to 1 pound is 1%. Assume that weight of coffee per can has a normal distribution with unknown mean and standard deviation.

Example - solution The decision rule is: Action 1 - adjust process if or if

Example - solution The decision rule is: Action 2 - do not adjust process if

Example - solution Suppose that for a given day and s = 0.012 Then t =

Example - solution so that - 2.797 < 2.5 < 2.797 and Action 2: no adjustment, is taken. We conclude that the true mean weight of coffee per can is 1 pound. We have thus tested the statistical hypothesis that  = 1 pound versus the alternative hypothesis that  does not equal 1 pound at the 1% level of significance.

Test of Means Let X1, …, Xn, be a random sample of size n, from a normal distribution with mean  and standard deviation , both unknown. To test the Null Hypothesis H0:  = 0 , a given or specified value against the appropriate Alternative Hypothesis 1. HA:  < 0 , or 2. HA:  > 0 , 3. HA:   0 ,

Test of Means 1. t < -t, n-1 , 2. t > t, n-1 , at the 100  % level of significance. Calculate the value of the test statistic Reject H0 if 1. t < -t, n-1 , 2. t > t, n-1 , 3. t < -t/2, n-1 , or if t > t/2, n-1 , depending on the Alternative Hypothesis. Test of Means

Test on Two Means Let X11, X12, …, X1n1 be a random sample of size n1 from N(m1, s1) and X21, X22, …, X2n2 be a random sample of size n2 from N(m2, s2), where m1, s1, m2 and s2 are all unknown. To test against the appropriate alternative hypothesis H0: µ1 - µ2 = do, where do  0 (usually do=0)

Test on Two Means 1. H1: µ1 - µ2 < do, where do  0, or at the a  100% level of significance, calculate the value of the test statistic.

Test on Two Means Calculate the value of the test statistic

Test on Two Means Reject Ho if 1. t' < -ta,ν or 2. t' > ta, ν or t' > ta/2, ν depending on the alternative hypothesis, where

Example - Test on Two Means An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested, by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave an average (coded) wear of 85 units with a standard deviation of 4, while samples of material 2 gave an average of 81 and a standard deviation of 5. Test the hypothesis that the two types of material exhibit the same mean abrasive wear at the 0.10 level of significance. Assume the populations to be approximately normal.

Example Test H0: m1 = m2 or m1 - m2 = 0. Vs. With a 10% level of significance, i.e., a = 0.10 Then

Example where and The calculate Critical Region is: t’ < -1.725 and t’ > 1.725,

Example Since t’ = 2.07, we can reject H0 and conclude that the two materials do not exhibit the same abrasive wear.

Test of Variances Let X1, …, Xn, be a random sample of size n, from a normal distribution with mean  and standard deviation , both unknown. To test the Null Hypothesis H0: 2 = o2 , a specified value against the appropriate Alternative Hypothesis 1. HA: 2 < o2 , or 2. HA: 2 > o2 , 3. HA: 2  o2 ,

Test of Variances at the 100  % level of significance. Calculate the value of the test statistic Reject H0 if 1. 2 < 21-, n-1 , 2. 2 > 2, n-1 , 3. 2 < 21-/2, n-1 , or if 2 > 2/2, n-1 , depending on the Alternative Hypothesis.

Test on Two Variances To test H0: Let X11, X12, …, X1n1 be a random sample of size n1 from N(m1, s1) and X21, X22, …, X2n2 be a random sample of size n2 from N(m2, s2), where m1, s1, m2 and s2 are all unknown. To test H0: against the appropriate alternative hypothesis

Test on Two Variances 1. H1: or 2. H1: 3. H1: at the a  100% level of significance, calculate the value of the test statistic.

Test on Two Variances Reject Ho if or depending on the alternative hypothesis, and where and

Example - Test on Variances An experiment was performed to compare the abrasive wear of two different laminated materials. Twelve pieces of material 1 were tested, by exposing each piece to a machine measuring wear. Ten pieces of material 2 were similarly tested. In each case, the depth of wear was observed. The samples of material 1 gave an average (coded) wear of 85 units with a standard deviation of 4, while samples of material 2 gave an average of 81 and a standard deviation of 5. Test the hypothesis that the two types of material exhibit the same variation in abrasive wear at the 0.10 level of significance.

Example - Test of Variances H0: s12 = s22 H1: s12  s22 With a 10% level of significance, i.e., a = 0.10 Critical region: From the graph we see that F0.05(11,9) = 3.11 0.34 0.05 3.11 x v1 = 11 and v2 = 9 f (x)

Example - Test of Variances Therefore, the null hypothesis is rejected when F < 0.34 or F > 3.11. Decision: Do not reject H0. Conclude that there is insufficient evidence that the variances differ.