More examples for Hypothesis Testing

Transcription

1 More example for Hypothei Tetig Part I: Compoet 1. Null ad alterative hypothee a. The ull hypothee (H 0 ) i a tatemet that the value of a populatio parameter (mea) i equal to ome claimed value. Ex H 0: µ=98.6 b. The alterative hypothee (H 1 or Ha) i the tatemet that the parameter ha a value that omehow differ for the ull hypothei. For the method of thi coure, the ymbolic form of the alterative hypothei mut ue oe of thee ymbol: < or > for directioal tet, or for o-directioal tet. 2. Directioality No-directioal tet (two-tailed tet) Ha: 0 (oe ample) 1 2 (two ample) Example keyword: compare, chage, differece, etc Directioal tet (oe-tailed tet) Ha: ( or ) 0 (oe ample) 1 ( or ) 2 (two ample) Example keyword: reduce, improve, delay, raie, higher, lower, greater tha, etc. 3. Deciio Truth Table1. Summary of Deciio ad Error Actio reject H 0 do ot reject H 0 H 0 i true type I error correct o-rejectio fale poitive H 0 i fale correct rejectio true poitive true egative type II error fale egative 4.. Tet tatitic The tet tatitic i a value computed from the ample data, ad it i ued i makig the deciio about the rejectio of the ull hypothei. The tet tatitic i foud by covertig the ample tatitic (uch a the ample mea) to a core (uch a t) with the aumptio that the ull hypothei i true. The tet tatitic ca therefore be ued for determiig whether there i igificat evidece agait the ull y hypothei. Tet tatitic take differet form. The baic form i t = 0. See table2 for more detail. 5. Sigificace level (deoted by α). Baelie of rejectio probability; Probability that the tet tatitic will fall i the critical regio whe the ull hypothei i actually true. If the tet tatitic fall i the critical regio, we will reject the ull hypothei, o α i the probability of makig the mitake of rejectig the ull hypothei whe it i true. y 6. Critical regio The critical regio (or rejectio regio) i the et of all value of the tet tatitic that caue u to reject the ull hypothei

2 7. Critical value A POSITIVE value that decide that boudary of the critical regio. The critical value chage depedig o the directioality. 8. P-value The probability of gettig a value of the tet tatitic that i at leat a extreme a the oe repreetig the ample data, aumig that the ull hypothei i true. Table2. Summary of Hypothei Compoet T-tet Oe Sample Two Idepedet Sample Null Alterative Hypothei Hypothei o-directioal directioal H 0 H A rejectio regio H A rejectio regio df t (1-) CI = 0 0 t < t /2 < 0 t < t -1 y for : 0 t > t /2 y t /2 > y 0 t > t 1-2 = t < t /2 t > t /2 1-2 < 0 t < t * 1-2 > 0 t > t y y1 y2 for : y1 y2 y 1 y 2 t /2 y1-y2 * df <

3 Part II: Procedure ad Example Thee tep are to be followed every time you do a hypothei tet. Step 1. State the cietific quetio to be awered. 2. Defie the parameter of iteret (Let be ) 3. State the ull hypothei H 0 ad H A mathematically (i term of parameter). 4. Name the type of tatitical tet to be ued ad the ditributio the tet tatitic follow uder H State the igificace level to be ued ad the correpodig critical value of the tet tatitic. Defie the rejectio regio. 6. Calculate the tet tatitic from the data. 7. Compare the tet tatitic to the rejectio regio, or, give the P-value ad compare it to. 8. Make a deciio about the ull hypothei: a) If the tet tatitic i i the rejectio regio, or the p-value i maller tha, tate "reject H 0 ". b) If ot, tate "do ot reject H 0 ". 9. Form a cietific cocluio baed o that deciio. If 8a) the tart with "Thi tudy provide evidece " if 8b) the tart with "Thi tudy doe ot provide evidece" followed by "at the igificace level that" followed by the verbal tatemet of H A. Remark Ue a complete etece with o ymbol. Thi ca be fairly geeral., or 1 ad 2, or, or 1 ad 2, or Uually a equality. Occaioally (e.g. F-tet, WMW) it may be too cumberome to expre H A mathematically. Iclude the df (or other parameter) for that ditributio if appropriate. The critical value ad rejectio regio deped o, the df ad ditributio from 5., ad whether H A i directioal. Show all calculatio. I.e. tet tatitic < or > critical value? If uig P-value, reject H 0 whe P <. Ue a complete etece with o ymbol. Ue imilar wordig to tep 4. You MUST NOT affirm H 0. See example. Do't get too creative.

4 Example: "oe-ample, o-directioal t-tet" I a tudy of the effect of alumium itake o the metal developmet of ifat, a group of 92 ifat who had bee bor prematurely were give a pecial alumium-depleted itraveoufeedig olutio. At age 18 moth the eurologic developmet of the ifat wa meaured uig the Bayley Metal Developmet Idex, which i deiged o that 100 i the average core i the geeral populatio. How do premature ifat fed thi olutio compare i eurologic developmet to the geeral populatio? BMDI core mea Solutio [ote i quare bracket are for your iformatio ad are ot part of the olutio] Do premature ifat fed a pecial alumium-depleted itraveou-feedig olutio have differet eurological developmet tha the geeral ifat populatio? Let be the mea BMDI core for premature ifat fed the olutio. Let 0 = 100; the mea BMDI core for the geeral ifat populatio. H 0 : = 0 ; premature ifat fed the olutio have the ame mea BMDI core a the geeral populatio. H A : 1 2 ; premature ifat fed the olutio have a differet mea BMDI core tha the geeral populatio. Ue a oe-ample, o-directioal t-tet. y t = 0 ha a t-ditributio with df = 1 = 92 1 = 91 uder H 0. Tet at igificace level = 0.05; critical value i t.025 = [Ue = 0.05 ule a differet level i pecified i the problem.] If t < or t > the will reject H = = y t = = < t < o do ot reject H 0 at level [ If t were < t.025 or > t.025 the would reject H 0.] Thi tudy doe ot provide evidece at the.05 igificace level that premature ifat fed the olutio have a differet mea BMDI core tha the geeral populatio. [If H 0 wa ot rejected, coclude "Thi tudy provide evidece at the.05 igificace level that premature ifat fed the olutio have a differet mea BMDI core tha the geeral populatio."] [If you wih, you may alo iclude the P-value i the cocludig etece. Place it i parethee (P= ) immediately before the word "at", i.e. "Thi tudy doe ot provide evidece (P= ) at the.05 igificace level that premature ifat fed the olutio have a differet mea BMDI core tha the geeral populatio." or "Thi tudy provide evidece (P = 0. ) at the.05 igificace level that premature ifat fed the olutio have a differet mea BMDI core tha the geeral populatio "]

5 Example: oe-ample, directioal t-tet I a tudy of a viral dieae, reearcher wih to kow if the viral ifectio elevate body temperature. We kow that the mea healthy body temperature i the geeral populatio i 37.0 C. A group of 25 patiet ufferig from thi viru have their temperature take. We oberve that their mea temperature i 37.5 C ad their SD i 1.0 C. Aumig that body temperature follow approximately a ormal ditributio, ue thee data to determie whether the viral ifectio elevate body temperature. Solutio [ote i quare bracket are for your iformatio ad are ot part of the olutio] Do people with thi viral ifectio have elevated body temperature? Let be the mea body temperature of all people ifected with thi viru. Let 0 = 37.0; the mea healthy body temperature i the geeral populatio H 0 : = 0 ; people ifected with thi viru have ormal mea body temperature H A : > 0 ; people ifected with thi viru have elevated mea body temperature Ue a oe-ample, directioal t-tet y t = 0 ha a t-ditributio with 1 = 24 df Tet at igificace level = 0.05; critical value i t.05 = [Ue = 0.05 ule a differet level i pecified i the problem.] If t > the will reject H 0. [Note: if H A wa "<" the thi tatemet would be "If t < the will reject H 0 "] 1.0 = = y t = 0 = ( )/0.2 = 2.5 t > o reject H 0 at level 0.05 [If t < the would ot reject H 0.] Thi tudy provide evidece at the 0.05 igificace level that people ifected with thi viru have elevated mea body temperature. [If H 0 wa ot rejected, cocluio would be "Thi tudy doe ot provide evidece at the 0.05 igificace level that people ifected with thi viru have elevated mea body temperature."]

6 Example: "two-idepedet-ample, o-directioal t-tet" I a tudy of the periodical cicada (Magicicada eptedecim), reearcher meaured the hid tibia legth of the hed ki of 100 idividual. Reult for male ad female are how i the accompayig table. Aumig that hid tibia legth follow a ormal ditributio, compare the mea hid tibia legth for male ad female cicade uig a appropriate hypothei tet. Tibia Legth (micrometre uit) Group mea SD Male Female Solutio [ote i quare bracket are for your iformatio ad are ot part of the olutio] Do male ad female cicada have the ame mea tibia legth? Let 1 be the mea tibia legth for male cicada ad 2 be the mea tibia legth for female cicada. H 0 : 1 = 2 ; Male ad female cicada have the ame mea tibia legth. H A : 1 2 ; Male ad female cicada have differet mea tibia legth. Ue a two-idepedet-ample, o-directioal t-tet. y y t = ha a t-ditributio with df = = 91 uder H 0. Tet at igificace level = 0.05; critical value i t.025 = [Ue = 0.05 ule a differet level i pecified i the problem.] If t < or t > the will reject H 0. (U) y1-y2 = = y1 y t = = t < o reject H 0 at level [ If t were > would alo reject H 0. If < t < would ot reject H 0.] Thi tudy provide evidece at the.05 igificace level that male ad female cicada have differet mea tibia legth. [If H 0 wa ot rejected, coclude "Thi tudy doe ot provide evidece at the.05 igificace level that male ad female cicada have differet mea tibia legth."] [If you wih, you may alo iclude the P-value i the cocludig etece. Place it i parethee (P= ) immediately before the word "at", i.e. "Thi tudy provide evidece (P= ) at the.05 igificace level that male ad female cicada have differet mea tibia legth." or, i the cae of o-rejectio (if the p-value i 0.6), "Thi tudy doe ot provide evidece (P= ) at the.05 igificace level that male ad female cicada have differet mea tibia legth."]

7 Example: "two-idepedet-ample, directioal t-tet" A pai-killig drug wa teted for efficacy i 50 wome who were experiecig uterie crampig pai followig childbirth. Twety-five of the wome were radomly allocated to receive the drug, ad the remaiig 25 received a placebo (iert ubtace). Capule of drug or placebo were give before breakfat ad agai at oo. A pai relief core, baed o hourly quetioig throughout the day, wa computed for each woma. The poible pai relief core raged from 0 (o relief) to 56 (complete relief for 8 hour). Summary reult are how i the table. Aumig that pai relief core follow approximately a ormal ditributio, tet for efficacy of the drug at reducig uterie crampig pai. PAIN RELIEF SCORE Treatmet Mea SD Placebo Drug Solutio I the drug more effective tha the placebo at reducig pai? Let 1 be the mea pai relief core of wome who take the placebo. Let 2 be the mea pai relief core of wome who take the drug. H 0 : 1 = 2 ; The drug ad the placebo are equivalet at reducig uterie crampig pai. [or, " The mea pai relief core of wome who take the drug i the ame a that for wome who take the placebo."] H A : 1 < 2 ; The drug i more effective tha the placebo at reducig uterie crampig pai. [or "The mea pai relief core of wome who take the placebo i le tha that of wome who take the drug." Ue a two-ample, directioal t-tet t = (y 1 - y 2 ) / ha a t-ditributio with df = = 47 uder H 0. Tet at igificace level = 0.05; critical value i t.05 = [Ue = 0.05 ule a differet level i pecified i the problem.] If t < the will reject H 0. [Note: if H A were ">" the thi tatemet would be "If t > the will reject H 0 "] (U) y1-y2 = = 3.66 y1 y t = = t > o do ot reject H 0 at level 0.05 [If t < the would reject H 0.] Thi tudy doe ot provide evidece at the 0.05 igificace level that the drug i more effective tha the placebo at reducig uterie crampig pai. [If H 0 wa rejected, cocluio would be "Thi tudy provide evidece at the 0.05 igificace level that the drug i more effective tha the placebo at reducig uterie crampig pai."] ]