STAT-UB.0103 NOTES for Wednesday 2012.APR.25. Here s a rehash on the p-value notion:

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1 STAT-UB.3 NOTES for Wedesday 22.APR.25 Here s a rehash o the -value otio: The -value is the smallest α at which H would have bee rejected, with these data. The -value is a measure of SHOCK i the data. Small s are very shockig. Here s a summary: >.5 NS ot sigificat give it u.<<.5 * sigificat iterestig.<<. ** highly sigificat excitig! <. ***very highly sigificat what a fid!!! If you MUST actually make a decisio (to urchase a ew set of cash registers, to adot a ew auditig rocedure, to iitiate a lawsuit, the you should try to set α o a ratioal basis. Thik of the costs of the errors! If you do t really have to make a decisio, as i a exloratory study, the the -value becomes a useful measure of the iterestigess of the results. This is exactly the way that Miitab uses the -value. There are a umber of very stadard statistical stories. Thik of these as fillig i a chart like this: Situatio (model Assumtios Radom variables Usual hyotheses Test method Cofidece iterval

2 This is the start for what we did recetly, alog with what we ll do today. Situatio (model Samle from oulatio with mea µ ad stadard deviatio σ Sigle biomial (, Assumtios Hoest samlig; either ormal oulatio or 3 Hoest samlig Radom variables X, X 2,, X ; the X, s. X evets i trials; the X Usual hyotheses H : µ µ H : µ µ H : H : t Test method X s µ ; reject H if t t α/2 ; - Z ˆ ( reject H if Z z α/2 ; for µ Cofidece iterval X ± tα/2; s for May choices, Agresti-Coull recommeded A full versio of this was distributed searately. 2

3 Now let s deal with the biomial radom variable X with ad. I geeral we do t kow, so we use the estimate X. This is read hat. We also oted that SD( stadard deviatio of the samlig distributio of SE( stadard error of estimate of the stadard deviatio of the samlig distributio of ˆ ( ˆ We had a umber of choices for the cofidece iterval for. The covetioal -α cofidece iterval is ± z SE( ˆ ( ˆ ˆ ± zα/2. that i this course, but it s quite legitimate. The cotiuity corrected iterval is ± z SE( ˆ α / 2. ˆ α / 2, meaig ( + 2. We did ot use ( You ll also see the form ± zα/2, with + 4 called Agresti-Coull, ad it s the oe we recommed. x This form is Now let s cosider a test of the ull hyothesis H :, where is some secified comariso value. The alterative will be H :. (If you like oe-sided tests, the you ca modify all this stuff i the obvious way. We ll say more about oe-sided tests ( later. If H is true, the the SD of is. This is ot a stadard error, it s a exact stadard deviatio (assumig that H is correct. If H holds, we do t have to estimate SD(. Thus it follows, if the samle size is reasoably large, that the ˆ ˆ distributio of Z ( ˆ is aroximately SD ˆ ( stadard ormal. This leads to the test based o Z. ( 3

4 EXAMPLE: A lumber millig rocess traditioally eds u with 28% of its laks i choice coditio, meaig suitable for furiture rather tha geeral costructio. A chage i the millig rocedure was tried for a batch of 6 laks, ad this roduced 26 i choice coditio. Is this a sigificat chage? SOLUTION: Ste : H :.28 versus H :.28 We would have to do some heavy iterretig to make this a oe-sided roblem. Let s just leave it as a o-cotroversial two-sided test. Ste 2: α.5. We were give o guidace o α, so let s use α.5. Ste 3: The test statistic will be Z ˆ (.28. The ull hyothesis will be rejected if Z z α/2 z Otherwise, we ll accet H or reserve judgmet. Ste 4: Fid that 26 6 ˆ ( The the umerical value of Z is where we will use Z ( Ste 5: Clearly H must be rejected. It looks like this ew millig rocedure roduces a higher roortio of good boards. You ca get Miitab to do the work (ot that this was difficult. Do Stat Basic Statistics Proortio Summarized Data. Be sure to use also Otios Use test ad iterval based o ormal distributio. This is the result: Test ad CI for Oe Proortio Test of.28 vs ot.28 Samle X N Samle 95% CI Z-Value P-Value (.37948,

5 If you do t check off the ormal distributio choice, the test ad cofidece iterval will be doe by aother method. Commets: The cofidece iterval that s give here is the covetioal oe, meaig ( ˆ ˆ ˆ ± zα/2. If you wated the Agresti-Coull iterval, just tell Miitab that you had 28 ( x + 2 evets i 64 ( + 4 trials, ad the ask for the test ad iterval based o the ormal distributio. You should robably ot use the test based o x + 2 evets i + 4 trials; that just raises too may questios. You will sometimes see this test i the form Z ˆ ˆ ( ˆ ˆ ˆ ( ˆ. This is ot the usual form, ad it s ot recommeded. For this examle, we d have Z , which is ot quite close eough! If you make the test usig Z the there is a exact equivalece betwee the covetioal cofidece iterval ad this Z versio of the hyothesis test. This ot a good eough argumet for usig Z. If you make the usual test usig Z, the accetig H is ot quite equivalet to havig i the cofidece iterval ± z α/2 SE(. How come? The cofidece iterval uses ˆ ( ˆ while the test uses (. It usually haes that ˆ ( ˆ (, so that there are oly rare stories i which the hyothesis test ad the cofidece iterval disagree. It s also ot quite equivalet to havig i the Agresti-Coull cofidece iterval. There is a cotiuity-corrected form i which the umerator is ˆ ; 2 that is, the calculatio is moved closer to by 2. 5

6 The requiremet for usig the ormal aroximatio is ot simly that is large. The requiremet is that 5 ad ( - 5. That is, whe H is true, the exected umber of successes ad the exected umber of failures both have to be at least 5. Some eole use oe-sided tests. It s a dagerous idea. Please see ages 3-8 of the hyothesis testig amhlet. As a fial commet o oe-sided tests... * I geeral, your motives may be questioed whe you do a oe-sided test, so geerally stay away from them. * There are some established frameworks i which oe-sided tests are doe all the time, or at least there is a stated traditio. These iclude testig for ollutio limits or checkig for accoutig overstatemets. Let s cosider aother roblem. Here s aother stadard cofidece iterval ad test. Cosider two samles: X, X 2,..., X m ad Y, Y 2,..., Y Here we ve oted the radom variables ad X s ad Y s. You ll see this also idetified as grou ad grou 2 or erhas grou A ad grou B. The usual assumtio is that these come from ormal oulatios with the same stadard deviatio σ but with (ossibly differet meas µ x ad µ y. We ll ow give the test of the obvious hyothesis testig roblem H : µ x µ y versus H : µ x µ y. I assig, we will ote also the cofidece iterval for the differece of two meas, meaig µ x - µ y. Begi by comutig s ooled stadard deviatio estimate, where s m bxi xg + cyj yh 2 i j m S xx + S yy m + 2 ( + ( m s s 2 2 x y m + 2 The value of s will lie betwee s x ad s y. two cometig estimates: Why do we do this? I this case we have 6

7 s x estimates σ from the X grou, ad it has m degrees of freedom s y estimates σ from the Y grou, ad it has degrees of freedom The statistic s is the recommeded way to combie (or ool this iformatio The test is based o the t statistic t m X Y m + s Whe H is true, this has the t distributio with m+-2 degrees of freedom. The rule the is to reject H whe t t α/2; m+-2. 7