AP Statistics Section 13.2 B

An observed difference between two sample proportions can reflect a difference in the

populations or it may just be due to chance variation in random sampling. Significance tests

help us decide if the effect we see in the samples is really there in the populations. The null hypothesis says that _________________

between the two populations: . The alternative hypothesis says what kind of

difference we expect.

difference no is there:0H 021 pp

In a significance test, we always assume___________________. So, if we assume there is no difference in the populations, then all

the observations in both samples really come from _______________. So instead of

estimating separately, we combine the two samples and use the overall sample

proportion to estimate the single population parameter p.

trueis hypothesis null the

population same the

21 p and p

This is called the combined sample proportion. It is

21

21

combined samplesboth in sindividual of #

combined samplesboth in successes of #ˆ

nn

xxpc

Using in place of both in the expression for the standard error

SE of gives

cp̂

21 p̂ and p̂

21 p and p

21

) )ˆ1(ˆˆ1(ˆ

n

pp

n

ppcccc )

11)(ˆ1(ˆ

21 nnpp cc

21

21

11)ˆ1(ˆ

ˆˆ

nnpp

pp

cc

10 all )ˆ1(,ˆ),ˆ1(,ˆ 2211 cccc pnpnpnpn

Example 13.12: The movie A Civil Action tells the following story. A town well that supplied water to East Woburn,

Massachusetts residents was contaminated by industrial chemicals. During the period that residents drank water from this well, a sample of 414 births showed 16 birth defects. On the west side of Woburn, a sample of 228

babies born during the same time period revealed 3 birth defects. The plaintiffs suing the companies responsible for the contamination claimed that these data show that the

rate of birth defects was significantly higher in East Woburn, where the contaminated well water was in use. How strong is the evidence supporting this claim? What

should the judge for this case conclude?

039.414

16ˆ Ep 013.

228

3ˆ Wp 030.

642

19ˆ Cp

Hypothesis:

Conditions: SRS:

Normality:

Independence:

ed.contaminat was well the time theduringburn in West Wo

born babies and East Wournin born babies areinterest of spopulationMy

0:H 0: a0 WEWE ppppH

burnin West Wo defectsbirth with babies of proportionp

nEast Woburin defectsbirth with babies of proportion

W Ep

.population the togeneralizenot willresults

not If SRSs. weresamples assume not toreason No

Normal approx. ison distributi assume tosafe so 5, All

221.16228(.97) 6.84228(.03) 58.401414(.97) 42.12)03(.414

sample. ingcorrespond the times10at bemust population

eacht replacemen without sampling are weSince

Calculations:

Interpretation:

85.1

2281

4141

)97)(.03(.

013.039.

z

85.1

032. valueP

.026. of difference a with sproportion sample twogetting of chance

3.2% a is therern, West Wobuandn East Woburin defectsbirth with

born babies of sproportion in the difference no is thereAssuming

n.East Woburin greater wasdefectsbirth born with babies

of proportion that theevidence strong moderately provides This