COMPARING TWO POPULATION PROPORTIONS USING INDEPENDENT SAMPLES

Transcription

1 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS COMPARING TWO POPULATION PROPORTIONS USING INDEPENDENT SAMPLES EXAMPLE: The article Foragig Behavior i the Idia False Vampire Bat reported that 36 of 93 female bats i flight spet more tha 5 miutes i the air before locatig food. For male bats, 64 of 68 exceeded 5 miutes whe locatig food. Is there sufficiet evidece to idicate that the proportio of flights takig loger tha 5 miutes differs for the two sexes? Note: two idepedet samples ad the iterest is i comparig the proportios for the two geders Notatio: Populatio Populatio Proportio Sample Size Sample Proportio To compare populatio proportios we usually cosider the size of the differece :

2 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 0 > 0 > < 0 < Our samplig estimate of this differece is the differece i the sample proportios whe the two samples are idepedet of oe aother. The estimator of is. Samplig Distributio of whe the two samples are idepedetly ad radomly take: ) the mea of the distributio is µ (that is, is ubiased) ) the stadard deviatio of the distributio is ( ) ( ) σ 3) the shape of the distributio is approximately ormal (a bell curve) if both ad are large. The sample sizes are usually cosidered to be sufficietly large if the followig statemet is true:

3 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 3 5, 5, ( ) 5 ( ) This, i words, says that you should have eough observatios from each populatio so that at least 5 observatios should be successes ad at least 5 should be failures. Hece, at a miimum, 0 ad 0 (but oly if 0. 5). 5 As we ll see, the estimator of σ ( ) ( ) depeds o whether we are costructig a cofidece iterval or performig a test of the differece i the two populatio proportios.

4 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 4 Large Sample Test of the Differece i Two Populatio Proportios Based o Two Idepedet Samples Null hypothesis: H 0 : 0 Alterative Hypothesis is oe of three: a) H A : > 0 b) H A : < 0 c) H A : 0 Test Statistic: z ( ) C ( C ) where C total # successes i both samples total sample size P-value: depeds o the alterative hypothesis: a) P-value Pr( Z > z) b) P-value Pr( Z < z) c) P-value Pr( Z < - z )

5 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 5 Decisio Rule: reject H o if P-value α Assumptios:. ad are large eough for the sample proportios to be approximately ormally distributed. the samplig was radom ad ot more tha 5% of the populatio. 3. the two samples are idepedetly take EXAMPLE Bats: Sample Statistics: Popula tio Sample Size #Successes female male Sample Proportio Hypotheses: H o : 0 H A : 0 Assumptios: 5, 5, ( ) 5 ( ) 5 have

6 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 6 bee met. Ad we have radom samples. Sigificace level: let s use α Test Statistic: first we eed the commo proportio C The, z ( ) C ( C ) ( ).77(.77) P-value: Pr(Z< - z ) Pr(Z<-4.) < Coclusios: We reject the ull hypothesis sice p- value <0.000 <<<< α0.05. There is strog evidece based o these samples, that the populatio proportio of female false vampire bats who take

7 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 7 loger tha 5 miutes searchig for food is differet from the proportio for male bats. EXAMPLE Old Faithful, the geyser at Yellowstoe Natioal Park, is kow to have two distict types of eruptios: log-duratio (> 3 miutes) ad short duratio (< 3 mi). If the types of eruptios are equally likely at all times of the day, the the proportio of log duratio eruptios occurrig durig the day should be the same as the proportio at ight. A geologist hypothesized that the legth of duratio was affected by solar heatig durig the day ad hece, the proportio of daytime log duratio eruptios should be higher tha the ight-time proportio. Two samples were take i August over several days ad ights. The geologist observed 53% log duratio eruptios durig the day (out of 35 eruptios) ad 49% (out of 4 eruptios) at ight. Is there sufficiet evidece to support the scietist s claim? Use a sigificace level of Hypotheses: H o : 0 H A : 0 > (populatio is the daytime eruptios ad, the ight time)

8 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 8 Assumptios: 5 ) ( 5, 5 ) ( 5,? radom samples? Sigificace level: α Test Statistic: first we eed the commo proportio The, ) ( ) ( z C C P-value:

9 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 9 Coclusios: Large Sample Cofidece Iterval Estimatio of The Differece Betwee Two Proportios Based o Idepedet Samples: Iterval Estimator: ( ) ± z α ( ) ( ) where the z critical value is based o the cofidece level ( α) desired Assumptios:. ad are large eough for the sample proportios to be approximately ormally distributed. the samplig was radom 3. the two samples are idepedetly take Note that the estimator of SE( ) is differet tha the oe used i hypothesis testig!

10 COMPARING TWO POPULATION (OR TREATMENT) PROPORTIONS 0 EXAMPLE for the bats let s use a 90% C.I. to estimate the differece i proportios of time spet searchig for food betwee males ad females. Now, the z critical value for 90% is.645. So, a 90% C.I. is ( ) ±.645 (.87.38) ± ±.645(.0468).94 ±.077 ( ).87(.87) 93 (.7,.7) ( ).38(.38) 68 Hece, with 90% cofidece, the populatio proportio of female false vampire bats that sped more tha 5 miutes locatig food is betwee.7% ad 7.% lower tha the populatio proportio of male bats which sped more tha 5 miutes locatig food. (We could reverse that ad say that the proportio of males spedig more tha 5 miutes is betwee.7 ad 7.% higher tha the proportio of females.)