Chapter 2 Hypothesis Testing

Chapter 2 Hypothesis Testing - 7.22 Are the data consistent...

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Unformatted text preview: 7.22. Are the data consistent with the assumed process mean? ...ENG|NEER|NG STATISTICS HANDBOOK. é TOOLS £- thS lSEARCH iBACK NEXTI 7.2.2. Are the data consistent with the assumed process mean? The testing of H0 for a single population mean Typical null hypotheses Test statistic where the standard deviation is not known Given a random sample of measurements. Y1. YN. there are three types of questions regarding the true mean of the population that can be addressed with the sample data. They are: 1. Does the true mean agree with a known standard or assumed mean? 2. Is the true mean of the population less than a given standard? 3. Is the true mean of the population at least as large as a given standard? The corresponding null hypotheses that test the true mean. r“, against the standard or assumed mean, ,I-l'g are: 1. fiozrn=rro 2. Howie!) 3 Hornzm The basic statistics for the test are the sample mean and the standard deviation. The form of the test statistic depends on whether the poulation standard deviation, or, is known or is estimated from the data at hand. The more typical case is where the standard deviation must be estimated from the data. and the test statistic is fro six/P7 I where the sample mean is _ 1 N r = — 2}:- N. 1:1 http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm Page 1 of 3 1/28/2012 Comparison with critical values Test statistic where the standard deviation is known Caution An illustrative example of the t-test The test is 7.22. Are the data consistent with the assumed process mean? and the sample standard deviation is with N - 1 degrees of freedom. For a test at significance level a, where a is chosen to be small, typically .01, .05 or .10, the hypothesis associated with each case enumerated above is rejected if: 1* i‘i'li2 Ianew—1 2. Iaiafifl 3. t s —ia;fi,_1 where Ema; 3H is the upper M2 critical value from the 2‘ distribution with N—l degrees of freedom and similarly for cases (2) and (3). Critical values can be found in the t-inblc in Chapter 1. If the standard deviation is known, the form of the test statistic is Y—ao z=—— Jivfi For case (1). the test statistic is compared with Zmz, which is the upper Cir-"2. critical value liom the standard normal distribution, and similarly for cases (2) and (3). If the standard deviation is assumed known for the purpose of this test, this assumption should be checked by a test of h I othesis for the standard deviation. The following numbers are particle (contamination) counts for a sample of 10 semiconductor silicon wafers: 50 48 44 56 61 52 53 55 67 51 The mean = 53.7 counts and the standard deviation = 6.567 counts. Over a long run the process average for wafer particle counts http://www.itl.nist.gov/div898/handbook/prc/section2/prc22.htm Page 2 of 3 1/28/2012 7.2.2. Are the data consistent with the assumed process mean? two—sided Critical values Conclusion NIST SEMATECH has been 50 counts per wafer. and on the basis of the sample. we want to test whether a change has occurred. The null hypothesis that the process mean is 50 counts is tested against the alternative hypothesis that the process mean is not equal to 50 counts. The purpose of the two—sided alternative is to rule out a possible process change in either direction. For a significance level of a = .05. the chances of erroneously rejecting the null hypothesis when it is true are 5% or less. (For a review of hypothesis testing basics. see Chapter 1). Even though there is a history on this process. it has not been stable enough to justify the assumption that the standard deviation is known. Therefore. the appropriate test statistic is the r-statistic. Substituting the sample mean. sample standard deviation. and sample size into the formula tor the test statistic gives a value of t= 1.782 with degrees of freedom = N - I = 9. This value is tested against the upper critical value t0.025;,9 : 2-262 from the t-tuble where the critical value is found under the column labeled 0.025 for the probability of exceeding the critical value and in the row for 9 degrees of freedom. The critical value Cir-’2 is used instead of it because of the two- sided alternative (two-tailed test) which requires equal probabilities in each tail of the distribution that add to at. WEE? [Toms a mos" :55“!ch {"hicT "‘e”"'x7‘""i"] http://www.itl.nist.gov/div89S/handbook/prc/sectionZ/prc22.htm Page 3 of3 1/28/2012 1.3.5.2. Confidence Limits for the Mean Page 1 of 5 I I ENGINEEING STATISTICS HANDBOOK Wm m rim:- warm 1.3.5.2. Confidence Limits for the Mean Purpose: Confidence limits for the mean (Sncdecor and Cochran. 1989) are an Interval interval estimate for the mean. Interval estimates are often desirable Estimate for because the estimate of the mean varies from sample to sample. Instead Mean of a single estimate for the mean. a confidence interval generates a lower and upper limit for the mean. The interval estimate gives an indication of how much uncertainty there is in our estimate of the true mean. The narrower the interval, the more precise is our estimate. Confidence limits are expressed in terms of a confidence coefficient. Although the choice of confidence coefficient is somewhat arbitrary. in practice 90%. 95%. and 99% intervals are often used. with 95% being the most commonly used. As a technical note. a 95% confidence interval does not mean that there is a 95% probability that the interval contains the true mean. The interval Wag} givengsagmpleefiither containsthe true mean or -._o— __ ___,_ I._,—--|_- -_—_- it does not. Instead. the level of confidence is associatedwith the method of calculating the interval. The COnfidenCe coefficient is simply the proportion of samples of a given size that may be expected to contain the true mean. That is. for a 95% confidence interval, if many samples are collected and the confidence interval computed. in the long run about 95% of these intervals would contain the true mean. Definition: Confidence limits are defined as: Confidence _ Interval y i tmflfi_l)3/¢N where Y is the sample mean. 5 is the sample standard deviation. N is the sample size. or is the desired significance level. and threw—.1} is the u ) Jcr critical value of the 1 distribution with N— 1 degrees of freedom. Note that the confidence coefficient is l - r1. From the formula. it is clear that the width of the interval is controlled by two factors: 1. As N increases. the interval gets narrower from the ‘JN term. http://www.itl.nist.gov/div898/handbook/eda/section3/eda3 52.htm 1/28/2012 1.3.5.2. Confidence Limits for the Mean Page 2 of 5 That is. one way to obtain more precise estimates for the mean is to increase the sample size. 2.. The larger the sample standard deviation, the larger the confidence interval. This simply means that noisy data. i.e., data with a large standard deviation. are going to generate wider intervals than data with a smaller standard deviation. Definition: To test whether the population mean has a Specific value. #0. against the Hypothesis two-sided alternative that it does not have a value Ho. the confidence Test interval is converted to hypothesis—test form. The test is a one-sample {- test, and it is defined as: H0: :3 2 Ho Ha: a 5'5 #0 Test Statistic: T = [i7 -— gm)/(s/VIE) where I”, N, and a are defined as above. Significance u: The most commonly used value for u: is 0.05. Level: Critical Region: Reject the null hypothesis that the mean is a specified value. #0. if T <1 —t(a,-'2,.v_1) or T I? tfrxli2.N—l} Sample Dataplot generated the following output for a confidence interval from Output for the ZARR'13.DAT data set: Confidence Interval CONFIDENCE LIMITS FOR MEAN (2-SIDED) NUMBER OF OBSERVATIONS MEAN STANDARD DEVIATION STANDARD_DEV 0. 8881E—01J/ 0.163194OE-O2 ll || CONFIDENCE T T x SD(MEAN) LOWER UPPER VALUE l%) VALUE LIMIT LIMIT 50.000 0.676 i0.110279E-02 9.26036 9.26256 75.000 1.154 0.188294E-02 9.25958 9.26334 90.000 1.653 0.269718E-02 9.25876 9.26416 95.000 1.972 0.321862E-02 9.25824 9.26468 99.000 2.601 0.424534E-02 9.25721 9.26571 99.900 3.341: 0.545297E-02 9.25601 9.26691 http://www.itl.nistgov/div898/handbook/eda/section3/eda3 52 .htm 1/28/2012 1.3.5.2. Confidence Limits for the Mean Page 3 of 5 99.990 3.973 0.648365E—02 9.25498 9.26794 99.999 4.536 0.740309E-02 9.25406 9.26886 Interpretation The first few lines print the sample statistics used in calculating the of the Sample confidence interval. The table shows the confidence interval for several Output different significance levels. The first column lists the confidence level (which is 1 - fl“ expressed as a percent), the second column lists the t- value (i.e.. Harmer—1}), the third column lists the t-value times the standard error (the standard error is s/VN). the fourth column lists the lower confidence limit. and the fifth column lists the upper confidence limit. For example, for a 95% confidence interval, we go to the row identified by 95.000 in the first column and extract an interval of (9.25824, 9.26468) from the last two columns. Output from other statistical software may look somewhat different from the above output. Sample Dataplot generated the following output for a one-sample t—test from the Output for t ZA RR] SDAT data set: Test T TEST (1—SANRLE) MUO = 5.000000 NULL HYPOTHESIS UNDER TEST--MEAN MU = 5.000000 SAMPLE: NUMBER OF OBSERVATIONS = 195 MEAN = 9.261460 0.2278881E-01 0.1631940E-02 STANDARD DEVIATION STANDARD DEVIATION OF MEAN TEST: MEAN—M00 = 4.261460 T TEST STATISTIC VALUE = 2611.284 DEGREES OF FREEDOM = 194.0000 T TEST STATISTIC CDF VALUE = 1.000000 ALTERNATIVE- ALTERNATIVE- ALTERNATIVE- HYPOTHESIS HYPOTHESIS HYPOTHESIS ACCEPTANCE INTERVAL CONCLUSION MU <> 5.000000 (0.0.025) (0.975.1) ACCEPT MU < 5.000000 (0.0.05) REJECT MU > 5.000000 (0.95.1) ACCEPT Interpretation gsamfle We are testing the hypothesis that the population mean is 5. The output “PM is divided into three sections. 1. The first section prints the sample statistics used in the computation of the t-test. http://www.itl.nist.gov/div898/handbook/eda/section3/eda352 .htm 1/28/2012 1.3.5.2. Confidence Limits for the Mean Questions Related Techniques Case Study Sofnvare The second section prints the t~test statistic value, the degrees of freedom, and the cumulative distribution function cdl value of the t-test statistic. The t-test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance intervals printed in section three. For an upper one-tailed test. the alternative hypothesis acceptance interval is (l - (1,1), the alternative hypothesis acceptance interval for a lower one-tailed test is (0.0:), and the alternative hypothesis acceptance interval for a two-tailed test is (l - (Jr/2,1) or (0.05/2). Note that accepting the alternative hypothesis is equivalent to rejecting the null hypothesis. The third section prints the conclusions for a 95% test since this is the most common case. Results are given in terms of the alternative hypothesis for the two-tailed test and for the one-tailed test in both directions. The alternative hypothesis acceptance interval column is stated in terms of the cdf value printed in section two. The last column specifies whether the alternative hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the t-test statistic cdf value printed in section two. For example. for a significance level of 0.10. the corresponding alternative hypothesis acceptance intervals are (0.005) and (095.1). (0. 0.10), and (0.90.1). Output from other statistical software may look somewhat different from the above output. Confidence limits for the mean can be used to answer the following questions: 1. What is a reasonable estimate for the mean? 2. How much variability is there in the estimate of the mean? 3. Does a given target value fall within the confidence limits? Two—Sample T—Test Confidence intervals for other location estimators such as the median or mid-mean tend to be mathematically difficult or intractable. For these cases. confidence intervals can be obtained using the bootstrap... Heat 'llrjiw meter data. Confidence limits for the mean and one-sample t-tests are available in just about all general purpose statistical software programs. including Datap I o t. http://www.itl.nistgov/divS98/handbook/eda/section3/eda352.htm Page 4 of 5 1/28/2012 1.3.5.3. Two-Sample <i>t</i>—Test for Equal Means Page 1 of 4 __E_NGINERING STATISTICS HANDBOOK W I'sen—ncfi' m Tie—am I I3 EDA Technicues 1.3.5. uantitutive Techni ues 1.3.5.3.. Two-Sample t—Test for Equal Means Purpose: The two—sample t-test (Sucdccor and Cochran. 1989) is used to Test if two determine if two population means are equal. A common application of population this is to test if a new process or treatment is superior to a current means are process or treatment. aqua! There are several variations on this test. 1. The data may either be paired or not paired. By paired. we mean that there is a one-to-one correspondence between the values in the two samples. That is. ile. X2. ....XH and Y1. Y . . Y” are the two samples. then X . corresponds to Y 1.. For paired samples. the difference X1. - Y]. is usually calculated. For unpaired samples. the sample sizes for the two samples may or may not be equal. The formulas for paired data are somewhat simpler than the formulas for unpaired data. 2. The variances of the two samples may be assumed to be equal or unequal. Equal variances yields somewhat simpler formulas. although with computers this is no longer a significant issue. 3. In some applications. you may want to adOpt a new process or treatment only if it exceeds the current treatment by some threshold. In this case. we can state the null hypothesis in the form that the difference between the two populations means is equal to some constant (#1 — fig 2 (in) where the constant is the desired threshold. Definition The two sample t test for unpaired data is defined as: #11 : fltz H3: #1 Té #2 Test Statistic: T _ Y1 — Y3 — if Sir/N1 + where N1 and N2 are the sample sizes. 17. and E are the sample means. and and are the sample variances. http://mvw.itl.nist.gov/div898/handbook/eda/section3/eda3 53 .htm 1/28/2012 Page 2 of 4 t0: 17 —}7 T: 1 2 Spqulle + where Significance or. Level: Crltrcal RBJBCt the null hypothesis that the two means are equal 1f Region T (C —t(‘1lf2:1) or T 3‘" tram») where Hagan) is the critical value of the I distribution with .12 degrees of freedom where U __ (Si/N1 + Sfi/Nzlz (sf/Ntlzrlel — 1) + (SS/Nzlg/UE — 1) If equal variances are assumed, then tJ::.Afil+—JV§-—EZ Sample Dataplot generated the following output for the t test from the Output A UT(_)8.SB.DAT data set: T TEST (2-SAMPLE) NULL HYPOTHESIS UNDER TEST—-POPULATION MEANS MUl = MU2 SAMPLE 1: NUMBER OF OBSERVATIONS = 249 MEAN = 20 . 14458 STANDARD DEVIATION = 6. 414700 STANDARD DEVIATION OF MEAN = 0.4065151 SAMPLE 2: NUMBER OF OBSERVATIONS = 79 MEAN -= 30 . «fl 8 1 0 l http://www.itl.nist.gov/diVS98/handbook/eda/section3/eda3 5 3 .htm 1/28/2012 1.3.5.3. Two—Sample <i>t</i>-Test for Equal Means Page 3 of 4 STANDARD DEVIATION = 6 107710 STANDARD DEVIATION OF MEAN = 0.6871710 IF ASSUME SIGMA1 = SIGMA2: POOLED STANDARD DEVIATION = 6.342600 DIFFERENCE (DEL) IN MEANS = —10.33643 STANDARD DEVIATION OF DEL = 0.8190135 T TEST STATISTIC VALUE = -12 62059 DEGREES OF FREEDOM = 326.0000 T TEST STATISTIC CDF VALUE = 0.000000 IF NOT ASSUME SIGMAI = SIGMAZ: STANDARD DEVIATION SAMPLE 1 = 6.414700 STANDARD DEVIATION SAMPLE 2 = 6.107710 BARTLETT CDF VALUE = 0.402799 DIFFERENCE (DEL) IN MEANS = —10 33643 STANDARD DEVIATION OF DEL = 0.7984100 T TEST STATISTIC VALUE = —12.94627 EQUIVALENT DEG. OF FREEDOM = 136.8750 T TEST STATISTIC CDP VALUE = 0.000000 ALTERNATIVE— ALTERNATIVE— ALTERNATIVE— HYPOTHESIS HYPOTHESIS HYPOTHESIS ACCEPTANCE INTERVAL CONCLUSION M01 <> M02 (0.0.025) (0.975.l) ACCEPT MUl < MU2 (0.0.05) ACCEPT MU1 > MU2 (0.95.1) REJECT Interpretation ofsampge We are testing the hypothesis that the population mean is equal for the Output two samples. The output is divided into five sections. 1. The first section prints the sample statistics for sample one used in the computation of the Host. 2. The second section prints the sample statistics for sample two used in the computation of the t-test. 3. The third section prints the pooled standard deviation. the difference in the means, the t-test statistic value, the degrees of freedom, and the cumulative distribution function Cdfl value Of the I-tcst statistic under the assumption that the standard deviations are equal. The Host statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance intervals printed in section five. For an upper one— tailed test, the acceptance interval is (0.1 - or). the acceptance interval for a two-tailed test is (Cr/2. 1 - nun/2), and the acceptance interval for a lower one-tailed test is ((1.1). 4. The fourth section prints the pooled standard deviation. the difference in the means. the t—test statistic value. the degrees of freedom, and the emmilstive distribution function (cdl) value of the (—test statistic under the assumption that the standard http://www.it1.nist.gov/div898/handbook/eda/section3/eda3 53 .htm 1/28/2012 1.3.5.3. Two—Sample <i>t</i>-Test for Equal Means Questions Related Techniques Case Study Software NIST SEMATECH deviations are not equal. The t-test statistic cdf value is an alternative way of expressing the critical value. cdf value is compared to the acceptance intervals printed in section five. For an upper one-tailed test. the alternative hypothesis acceptance interval is (1 - [1.1). the alternative hypothesis acceptance interval for a lower one—tailed test is (0.:1). and the alternative hypothesis acceptance interval for a two-tailed test is (1 - (Jr/2.1) or (0.0/2). Note that accepting the alternative hypothesis is equivalent to rejecting the null hypothesis. The fifth section prints the conclusions for a 95% test under the assumption that the standard deviations are not equal since a 95% test is the most common case. Results are given in terms of the alternative hypothesis for the two-tailed test and for the one-tailed test in both directions. The alternative hypothesis acceptance interval column is stated in terms of the cdf value printed in section four. The last column specifies whether the alternative hypothesis is accepted or rejected. For a different significance level. the appropriate conclusion can be drawn from the Host statistic Cdf value printed in section four. For example. for a significance level of 0.10. the corresponding alternative hypothesis acceptance intervals are (0.005) and (095.1). (0. 0.10). and (090.1). Output from other statistical software may look somewhat different from the above output. Two-sample t-tests can be used to answer the following questions: 1. 2. 3. Is process 1 equivalent to process 2? Is the new process better than the current process? Is the new process better than the current process by at least some pre-determined threshold amount? Confidence Limits for the Mean Analvsis of Variance Ceramic strength data. Two-sample t—tests are available in just about all general purpose statistical software programs. including Dataplot. ,pfin-ww-m 1-”!- Wm H http ://www.itl.nist. gov/div898/handbook/eda/section3 /eda3 5 3 .htm Page 4 of4 1/28/2012 1.3.5.8. Chi-Square Test for the Standard Deviation Page 1 of 3 iTOOLS 3‘- AIDS 1$EAICH HACK NEXT! l. . |.3. EDA Techniques 1 3.5. Quantitative Techniques 1.3.5.8. Chi-Square Test for the Standard Deviation Purpose: A chi—square test ( Snedecor and C ochran. 1983) can be used to test if the T est if standard deviation of a population is equal to a specified value. This test can be standard either a two-sided test or a one-sided test. The two-sided version tests against deviation is the alternative that the true standard deviation is either less than or greater than equal to a the specified value. The one-sided version only tests in one direction. The specified choice of a two—sided or one—sided test is determined by the problem. For value example. if we are testing a new process. we may only be concerned if its variability is greater than the variability of the current process. Definition The chi-square hypothesis test is defined as: H0: 0 = 00 H3: 5 <5 Us for a lower one-tailed test IT 3" Us for an upper one-tailed test 0 E’é an for a two-tailed test Test T=(N — 1)[:3/Ja)2 Statistic: where N is the sample size and s is the sample standard deviation. The key element of this formula is the ratio 3/03 which compares the ratio of the sample standard deviation to the target standard deviation. The more this ratio deviates from 1, the more likely we are to reject the null hypothesis. Significance :1. Level: C 't' al _ _ _ _ _ Rzlglign: Re} ect the null hypotheSIS that the standard devtatlon IS a specified value, mu. if T L)» xii, N_ 1} for an upper one-tailed alternative T «c: Xfi_mN_n for a lower one-tailed alternative T <1 x3 _m;21N_1} for a two-tailed test 01' http://www.itl.nist.gov/divS98/handbook/eda/section3/eda358.htm 1 02/701 ’3 1.3.5.8. Chi-Square Test for the Standard Deviation Page 2 of 3 T > Xian—1} where xii-{N4} 1s the crlucal value of the Chi-St; uare d'lSll‘lbLllmn with N - 1 degrees of freedom. In the above formulas for the critical regions, the Handbook follows the convention that X: is the upper critical value from the chi-square distribution and Xf—n: is the lower critical value from the chi-square distribution. Note that this is the opposite of some texts and software programs. In particular, Dataplot uses the opposite convention. The formula for the hypothesis test can easily be converted to form an interval estimate for the standard deviation: Sample Dataplot generated the following output for a Chi-square test from the Output GEARDAT data set: CHITSQUARED TEST SIGMAO = 0.1000000 NULL HYPOTHESIS UNDER TEST--STANDARD DEVIATION SIGMA = .1000000 SAMPLE: NUMBER OF OBSERVATIONS MEAN STANDARD DEVIATION S 100 0.9976400 0.6278908E-02 ll 11 TEST: S/SIGMAO = 0.6278908E—01 CEI-SQUARED STATISTIC = 0 3903044 DEGREES OF FREEDOM = 99.00000 CHI—SQUARED COE VALUE = 0.000000 ALTERNATIVE* ALTERNATIvE~ ALTERNATIVE- HYPOTHESIS HYPOTHESIS HYPOTHESIS ACCEPTANCE INTERVAL CONCLUSION SIGMA <> .1000000 (0.0.025), (0.975.I) ACCERT SIGMA < .1000000 (0.0.05) ACCEPT SIGMA > .lOOOOOO (0.95.1) REJECT Interpretation ofSampZe We are testing the hypothesis that the population standard deviation is 0.1. The Output output is divided into three sections. 1. The first section prints the sample statistics used in the computation of the chi-square test. http ://www.itl.nist.gov/div898/handbook/eda/section3/eda3 58 .htm 1/30/2012 1.3.5.8. Chi-Square Test for the Standard Deviation Page 3 of 3 2. The second section prints the chi-square test statistic value, the degrees of freedom, and the cumulative distribution 'l'unction cdl value of the chi-s uare test statistic. The chi-square test statistic cdf value is an alternative way of expressing the critical value. This cdf value is compared to the acceptance intervals printed in section three. For an upper one-tailed test. the alternative hypothesis acceptance interval is (1 - (1,1), the alternative hypothesis acceptance interval for a lower one-tailed test is (0.0). and the alternative hypothesis acceptance interval for a two- tailed test is (1 - [Jr/2,1) or (0.11/2). Note that accepting the alternative hypothesis is equivalent to rejecting the null hypothesis. 3. The third section prints the conclusions for a 95% test since this is the most common case. Results are given in terms of the alternative hypothesis for the two-tailed test and for the one-tailed test in both directions. The alternative hypothesis acceptance interval column is stated in terms of the cdf value printed in section two. The last column specifies whether the alternative hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the chi-square test statistic cdf value printed in section two. For example. for a significance level of 0.10, the corresponding alternative hypothesis acceptance intervals are (0.0.05) and (0.95.1), (O. 0.10), and (0.90.1). Output from other statistical software may look somewhat different from the above output. Questions The chi-square test can be used to answer the following questions: 1. Is the standard deviation equal to some pre-determined threshold value? 2. Is the standard deviation greater than some pre-determined threshold value? 3. Is the standard deviation less than some pre-determined threshold value? Related Test / Techniques Bartlett "licst firflflgsy- l...evene Test SofMare The chi-square test for the standard deviation is available in many general purpose statistical software programs. including Dataplot. N 1 WWW ......... __.,_ W... W..- Mai-Em lHOME toms 3. mos {SEARCH lance; want 5 E M AT E C H http://www.itl.nistgov/divS98/handbook/eda/section3/eda3 5 8 .htm 1/28/2012 1.3.5.9. F—Test for Equality of Two Standard Deviations . Ex aloratorv Data Anal =sis l [.3. EDA Tochni nos 1 uantitative Technit 1.3.5.9. F—Test for Equality of Two Standard Deviations Purpose: Test if standard deviations from two populations are equal Definition An F-test ( Suede-cor and Cochran- 1983) is used to test if the standard deviations of two populations are equal. This test can be a two-tailed test or a one-tailed test. The two-tailed version tests against the alternative that the standard deviations are not equal. The one—tailed version only tests in one direction. that is the standard deviation from the first pOpulation is either greater than or less than (but not both) the second population standard deviation . The choice is determined by the problem. For example, if we are testing a new process. we may only be interested in knowing if the new process is less variable than the old process. The F hypothesis test is defined as: H0 01 I 02 H3: ‘71 "C ‘72 for a lower one tailed test 0 1 3’ t1'2 for an upper one tailed test ‘71 3'5 ‘72 for a two tailed test Test F = / Statlstlc population variances. Significance [1 Level: Critical Region: The hypothesis that the two standard deviations are equal is rejected if I" 32> Fta,N1_1,N2.—1) for an upper one-tailed test F <2 F[1_.;1’N1-—1,N2—1) for a lower one—tailed test Page 1 of3 1/28/2012 1.3.5.9. F-Test for Equality of Two Standard Deviations Page 2 of 3 f" 3‘" filIIIZM—IJVZ—ll where F[¢,fi_1,1v_fi} is the critical value of the E distribution with “I and be degrees of freedom and a significance level of or. In the above formulas for the critical regions, the Handbook follows the convention that F”. is the upper critical value from the F distribution and F141 is the lower critical value from the F distribution. Note that this is the opposite of the designation used by some texts and software programs. In particular. Dataplot uses the opposite convention. Sample Dataplot generated the following output for an F-test from the Output JAHAN MllDAT data set: F TEST NULL HYPOTHESIS UNDER TEST--SIGMA1 = STGMA2 ALTERNATIVE HYPOTHESTS UNDER TEST--SIGMA1 NOT EQUAL SIGMAZ SAMPLE 1: NUMBER OF OBSERVATIONS = 240 MEAN = 688.9987 STANDARD DEVIATION = 65.54909 SAMPLE 2: NUMBER OF OBSERVATIONS = 240 MEAN = 611.1559 STANDARD DEVIATION = 61.85425 TEST: STANDARD DEv. (NUMERATOR) = 65 54909 STANDARD DEv. (DENOMINATOR) = 61.85425 E TEST STATISTIC VALUE = 1.123037 DEG. OF FREEDOM (NUMER.) = 239.0000 DEG. OF FREEDOM (DENOM ) = 1239400001 . E TEST STATISTIC CDF VALUE = ;10.814808 ‘\ . -__ a _fl_,L- NULL NULL HYPOTHESIéi NULL HYPOTHESIS HYPOTHESIS ACCEPTANCE INTERVAL CONCLUSION SIGMAl = SIGMAZ (0.000.03950) ACCEPT Interpretation of Sample We are testing the hypothesis that the standard deviations for sample one Output and sample two are equal. The output is divided into four sections. 1. The first section prints the sample statistics for sample one used in the computation of the F-test. 2. The second section prints the sample statistics for sample two used in the computation of the F-test. http://www.itl.nist.gov/div898/handbook/eda/section3/eda3 59.htm 1/28/2012 1.3.5.9. F-Test for Equality of Two Standard Deviations Questions Related Techniques Case Study Sofmare HIST SEMATECH the cumulative distribution statistic. The F—test statistic cdf value is an alternative way of expressing the critical valuc. This cdf value is compared to the acceptance interval printed in section four. The acceptance interval for a two-tailed test is (0.1 - 0:). The fourth section prints the conclusions for a 95% test since this is the most common case. Results are printed for an upper one— tailed test. The acceptance interval column is stated in terms of the cdf value printed in section three. The last column specifies whether the null hypothesis is accepted or rejected. For a different significance level, the appropriate conclusion can be drawn from the F-test statistic cdf value printed in section four. For example, for a significance level of 0.10, the corresponding acceptance interval become (0.000,0.9000). Output from other statistical software may look somewhat different from the above output. The F—test can be used to answer the following questions: 1. 2. uantilc- Do two samples come from populations with equal standard deviations? Does a new process. treatment... or test reduce the variability of the current process? " t‘tantilc P lot Bihistogram Chi—Sq uare Test Bartlett's Test Levcnc il‘est Ceramic stren 3th data. The F-test for equality of two Standard deviations is available in many general purpose statistical software programs, including D2an lot. mom Wm M Warn-«mail; {Tics mam treats a. 'Alos‘ {SEMEF Page 3 of 3 1/28/2012 ...
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This note was uploaded on 02/02/2012 for the course EMA 3800 taught by Professor El-shall during the Spring '10 term at University of Florida.

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Chapter 2 Hypothesis Testing - 7.22 Are the data consistent...

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