+ 1 Section 7.2 Sample Proportions

Learning Objectives After this section, you should be able to…

 FIND the mean and standard deviation of the of a sample proportion

 DETERMINE whether or not it is appropriate to use the Normal approximation to calculate probabilities involving the sample proportion

 CALCULATE probabilities involving the sample proportion

 EVALUATE a claim about a population proportion using the sampling distribution of the sample proportion Consider the approximate sampling distributions sampling Consider theapproximate

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of the this

question. p ? p ˆ

+ 2 characteristics for shape, center, and spread: Youhas thefollowing should havenoticedthesamplingdistribution  The SamplingDistributionfortheStatistic sample approximat Shape

because thesample proportion Center

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is

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 p ˆ  p  ˆ p ˆ  X  / n 1 n 1 n ( np np then (1 )   p p ) X  are: As sample size increases, thespread decreases. p ˆ np  p ( ˆ (1 1

n is / THE STATISTIC  n

 2 an ) X  p unbiased X  ) np 

p estimator (1

n

  p X ) for  p p ˆ np and a (1  p )

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when the ( 1

  must p

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+ 6 As

of successes.Let Choose anSRSofsize

The The n 10 and the Normal calculations,checkthat you perform increases, the sampling distributionbecomes increases, the as longthe standard mean n (1 – the ofthesamplingdistribution of p

sampling ) deviation Sampling DistributionofaSample Proportion We ≥ 10. 10 p ˆ bethesampleproportionof successes.Then:

can

%  condition n summarize p ˆ

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facts

approximately Normal p Normal condition ˆ p ˆ

n

is

as  about  1/10) (1

follows p ˆ  N p N withproportion p ˆ .

is : is satisfied: . Before np p ≥

+ 7  Example 1: See next slide forworked out solution See

+ 8 + 9  Example 2: will give aresultwithin 2 percen What is sample of1500students the probability that therandom actually attendcollege within 50 miles year students of home. all first- of 35% that is. Suppose their home far away how students A polling organization asksanSRSof 1500 first-year college So what are they asking? So whatare they Draw apicture! tage points of this truevalue? points tage

+ 10 of thistruevalue? the probabilitythatrandomsample of1500studen first-year studentsactually of all Suppose that35% A1500 first-year of polling organizationasksanSRS  0.33 and 0.37 (within 2 percentage points, or 0.02, of 0.35). or 0.02, 2 percentage points, and 0.37 (within 0.33 Webetween that thesampleproportionfalls wanttofindtheprobability STATE: Example 2: attend college within 50 miles of home.Whatis 50 miles attendcollegewithin college students how f college studentshow ts give a result will within 2percentage points within 50 miles of home. miles of within 50 p in whichproportion the a population drawn from size SRS of PLAN = 0.35 attend college = 0.35attend : Wehave an ar awaytheirhome is. Keep Going! n = 1500

+ 11 CONCLUDE points ofthe truthabout the population.  •And both are bothgreaterthan10, •Since Can we usethenormalmodel? • Next standardizetofindthedesired probability. Example 2(Cont): np = 1500(0.35)525 and : About 90% of all SRSs of size 1500 will give a result within 2 percentage About SRSs give a result within 90% ofall of size1500 will Since we know p Since we can find the meanandstandarddeviation can find n we can use the normal model. normal we can usethe (1 – P (0.33 p ( z p ) =1500(0.65)=975    = 0.35) p ˆ p 0 ˆ 

.  33 0 0.35 . 0.37) 0123  0 . and n 35  P  (   1 1.63 . (n 63

= 1500)   Z p ˆ   1.63) : (0.35)(0.65) then we  z 0.9484 1500  0 . 37 0 . 0123   0 0.0516 . 35  0.0123  1  . 63 0.8968 + 12 percentage points of the true value? points of percentage thetrue of probability thatanSRS What isthe Example 3: in her district are planning four-year collegeoruniversity. Suppose planning proportion to attend district are of highschoolstudentsin a her The Superintendent of a large school wants The Superintendentofalarge to knowthe to attendafour-yearcollege oruniversity. size125 will givearesult within7 that 80%ofallhighschoolstudents See next slide forworked out solution See

+ 13 + 14 + 15 + 16 Sample Proportions

Summary In this section, we learned that… When we want information about the population proportion p of successes, we  often take an SRS and use the sample proportion pˆ to estimate the unknown parameter p. The sampling distribution of pˆ describes how the statistic varies in all possible samples from the population. The mean of the sampling distribution of pˆ is equal to the population proportion  p . That is, pˆ is an unbiased estimator of p. p(1  p) The standard deviation of the sampling distribution of pˆ is   for pˆ n an SRS of size n. This formula can be used if the population is at least 10 times as large as the sample (the 10% condition). The standard deviation of pˆ gets smaller as the sample size n gets larger.  When the sample size n is larger, the sampling distribution of pˆ is close to a p(1  p) with mean p and standard deviation  pˆ  . n  In practice, use this Normal approximation when both np ≥ 10 and n(1 - p) ≥ 10 (the Normal condition).