# Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test.

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1 HYPOTHESIS TESTING

2 Learning Objectives Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test. Know how to perform a hypothesis test to compare a sample statistic to a target value. Know how to interpret a hypothesis test.

3 How does it help? Hypothesis testing is is necessary to: determine when there is is a significant difference between two sample populations. determine whether there is is a significant difference between a sample population and a target value.

4 IMPROVEMENT ROADMAP Uses of Hypothesis Testing Common Uses Phase 1: Measurement Characterization Breakthrough Phase : Analysis Confirm sources of variation to determine causative factors (x). Strategy Optimization Phase 3: Improvement Phase 4: Control Demonstrate a statistically significant difference between baseline data and data taken after improvements were implemented.

5 KEYS TO SUCCESS Use hypothesis testing to explore the data Use existing data wherever possible Use the team s experience to direct the testing Trust but verify.hypothesis testing is the verify If there s any doubt, find a way to hypothesis test it

6 SO WHAT IS HYPOTHESIS TESTING? The theory of probability is nothing more than good sense confirmed by calculation. Laplace We think we see something well, we think err maybe it it is could be. But, how do we know for sure? Hypothesis testing is is the key by giving us a measure of how confident we can be in our decision.

7 SO HOW DOES THIS HYPOTHESIS STUFF WORK? Critical Point Toto, I don t think we re in Kansas anymore. H o = no difference (Null hypothesis) Statistic CALC < CALC Statistic CRIT CRIT H 1 = significant difference (Alternate Hypothesis) Statistic CALC > CALC Statistic CRIT CRIT Statistic Value We determine a critical value from a probability table for the statistic. This value is is compared with the calculated value we get from our data. If If the calculated value exceeds the critical value, the probability of of this occurrence happening due to to random variation is is less than our test α. α.

8 SO WHAT IS THIS NULL HYPOTHESIS? Mathematicians are like Frenchmen, whatever you say to them they translate into their own language and forth with it is something entirely different. Goethe Hypothesis Symbol How you say it What it means Null H o Fail to Reject the Null Hypothesis Alternative H 1 Reject the Null Hypothesis Data does not support conclusion that there is a significant difference Data supports conclusion that there is a significant difference

9 HYPOTHESIS TESTING ROADMAP... Population Average Population Variance Compare Population Averages Compare a Population Average Against a Target Value Compare Population Variances Compare a Population Variance Against a Target Value Test Used Test Used Test Used Test Used Z Stat (n>30) Z Stat (n>30) F Stat (n>30) F Stat (n>30) Z Stat (p) Z Stat (p) F Stat (n<10) χ Stat (n>5) t Stat (n<30) t Stat (n<30) χ Stat (n>5) τ Stat (n<10) τ Stat (n<10)

10 HYPOTHESIS TESTING PROCEDURE Determine the hypothesis to be tested (Ho:=, < or >). Determine whether this is a 1 tail (a) or tail (a/) test. Determine the a risk for the test (typically.05). Determine the appropriate test statistic. Determine the critical value from the appropriate test statistic table. Gather the data. Use the data to calculate the actual test statistic. Compare the calculated value with the critical value. If the calculated value is larger than the critical value, reject the null hypothesis with confidence of 1-a (ie there is little probability (p<a) that this event occurred purely due to random chance) otherwise, accept the null hypothesis (this event occurred due to random chance).

11 WHEN DO YOU USE a VS a/? Many test statistics use α and others use α/ and often it is confusing to tell which one to use. The answer is straightforward when you consider what the test is trying to accomplish. If there are two bounds (upper and lower), the α probability must be split between each bound. This results in a test statistic of α/. If there is only one direction which is under consideration, the error (probability) is concentrated in that direction. This results in a test statistic of α. α α/ Examples Critical Critical Value Value Critical Critical Value Value Critical Critical Value Value Examples Ho: μ 1 >μ 1 >μ Ho: σ 1 <σ 1 <σ Common Occurrence Rare Occurrence Rare Occurrence Common Occurrence Rare Occurrence Ho: μ 1 =μ 1 =μ Ho: σ 1 =σ 1 =σ 1 α Probability α Probability α/ Probability 1 α Probability α/ Probability Confidence Confidence Int Int Test Test Fails Fails in in one one direction = α Test Test Fails Fails in in either direction = α/ α/

12 Sample Average vs Sample Average Coming up with the calculated statistic... n1= n 0 n1 + n 30 n + n > 1 X X τ dcalc = R + R 1 t = 1 1 [( n 1) s1 + ( n 1) s] + n n n + n 1 1 Z CALC = 1 30 Sample t Sample Tau Sample Z (DF: n 1 +n -) X X 1 X σ1 n X σ1 + n 1 1 Use these formulas to calculate the actual statistic for comparison with the critical (table) statistic. Note that the only major determinate here is the sample sizes. It should make sense to utilize the simpler tests ( Sample Tau) wherever possible unless you have a statistical software package available or enjoy the challenge.

13 Hypothesis Testing Example ( Sample Tau) Several changes were made to the sales organization. The weekly number of orders were tracked both before and after the changes. Determine if the samples have equal means with 95% confidence. Ho: m 1 = m Statistic Summary: n 1 = n = 5 Significance: a/ =.05 ( tail) tau crit =.613 (From the table for a =.05 & n=5) Calculation: R 1 =337, R = 577 X 1 =868, X =896 tau CALC =( )/( )=.06 X X τ dcalc = R + R 1 1 Test: H o : tau CALC < tau CRIT H o :.06<.613 = true? (yes, therefore we will fail to reject the null hypothesis). Receipts 1 Receipts Conclusion: Fail to reject the null hypothesis (ie. The data does not support the conclusion that there is a significant difference)

14 Hypothesis Testing Example ( Sample t) Several changes were made to the sales organization. The weekly number of orders were tracked both before and after the changes. Determine if the samples have equal means with 95% confidence. Ho: m 1 = m Statistic Summary: n 1 = n = 5 DF=n 1 + n - = 8 Significance: a/ =.05 ( tail) t crit =.306 (From the table for a=.05 and 8 DF) Calculation: s 1 =130, s = 7 X 1 =868, X =896 t CALC =( )/.63*185=.4 Test: H o : t CALC < t CRIT t = 1 1 [( n 1) s1 + ( n 1) s + ] n n n + n 1 Receipts 1 Receipts H o :.4 <.306 = true? (yes, therefore we will fail to reject the null hypothesis). Conclusion: Fail to reject the null hypothesis (ie. The data does not support the conclusion that there is a significant difference X X 1 1

15 Sample Average vs Target (m 0 ) Coming up with the calculated statistic... n 0 n 30 n > 30 1 Sample Tau 1 Sample t (DF: n-1) 1 Sample Z = X μ X μ 0 X μ 0 0 t CALC = Z CALC = R s s n n τ 1CALC Use these formulas to calculate the actual statistic for comparison with the critical (table) statistic. Note that the only major determinate again here is the sample size. Here again, it should make sense to utilize the simpler test (1 Sample Tau) wherever possible unless you have a statistical software package available (minitab) or enjoy the pain.

16 Sample Variance vs Sample Variance (s ) Coming up with the calculated statistic... n 1 < 10, n < 10 n 1 > 30, n > 30 Range Test F = CALC R R MAX, n MIN, n 1 F Test (DF 1 : n 1-1, DF : n -1) F s s calc = MAX MIN Use these formulas to calculate the actual statistic for comparison with the critical (table) statistic. Note that the only major determinate again here is the sample size. Here again, it should make sense to utilize the simpler test (Range Test) wherever possible unless you have a statistical software package available.

17 Hypothesis Testing Example ( Sample Variance) Several changes were made to the sales organization. The number of receipts was gathered both before and after the changes. Determine if the samples have equal variance with 95% confidence. Ho: s 1 = s Statistic Summary: n 1 = n = 5 Significance: a/ =.05 ( tail) F crit = 3.5 (From the table for n 1, n =5) Calculation: R 1 =337, R = 577 F CALC =577/337=1.7 Test: H o : F CALC < F CRIT F = CALC H o : 1.7 < 3.5 = true? (yes, therefore we will fail to reject the null hypothesis). Conclusion: Fail to reject the null hypothesis (ie. can t say there is a significant difference) R R MAX, n MIN, n 1 Receipts 1 Receipts

18 HYPOTHESIS TESTING, PERCENT DEFECTIVE n 1 > 30, n > 30 Compare to target (p 0 ) Compare two populations (p 1 & p ) Z CALC = p p p 1 0 ( 1 p ) 0 0 n Z CALC = np n _ np + n p p 1 np 1 1_ np n + n n n 1 1 Use these formulas to calculate the actual statistic for comparison with the critical (table) statistic. Note that in both cases the individual samples should be greater than 30.

19 How about a manufacturing example? We have a process which we have determined has a critical characteristic which has a target value of.53. Any deviation from this value will sub-optimize the resulting product. We want to sample the process to see how close we are to this value with 95% confidence. We gather 0 data points (shown below). Perform a 1 sample t test on the data to see how well we are doing

20 Learning Objectives Understand the role that hypothesis testing plays in an improvement project. Know how to perform a two sample hypothesis test. Know how to perform a hypothesis test to compare a sample statistic to a target value. Know how to interpret a hypothesis test.

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