Sampling Distribution of Sample proportion

Sampling Distribution of Sample proportions

1. Sampling Distribution of
Sample Proportion
Unit 5
This is the last part of Unit 5

3. Sampling Distribution of Sample Proportion
Assumethatinanelectionracebetween
CandidateAandCandidateB,0.60ofthevoters
preferCandidateA.
Ifarandomsampleof10voterswerepolled,itis
unlikelythatexactly60%ofthem,or6,would
preferCandidateA.
Bychancetheproportioninthesamplepreferring
CandidateAcouldeasilybealittlelowerthan0.60
oralittlehigherthan0.60.

4. Sampling Distribution of Sample Proportion
Thesamplingdistributionofpisthe
distributionthatwouldresultifyou
repeatedlysampled10votersand
determinedtheproportionpthatfavored
CandidateA.Ofcourseitdidnothavetobe
10!10wasonlythesamplesizewhichwe
cancallnandNisthetotalnumberinthe
population.

5. Central Limit Theorem
According to the Central Limit
Theorem, the sampling distribution
of p-hat is approximately normal for
sufficiently large sample size, such
that np > 5 and nq > 5.

6. Calculations
The formulae for calculating these proportions are given as follows:
p = X/N and Ƹ??????= x/n
where:
◦X is the number of elements in the population possessing the
characteristic of interest; N is the total number of elements in the
population;
◦x is the number of elements in the sample possessing the characteristic
of interest; and n is the total number of elements in the sample.
The distribution of p is closely related to the Binomial
distribution.

7. Mean and Standard Deviation of p

8. Let us look at some examples…
On page 75 of your unit notes, there is Example 5.1:
We have five students A, B, C, D and E and we are interested
in knowing which of these took statistics in sixth form. It
turns out that A , D and E did. The others not.
Based on this information, we have the proportion of
students who took statistics in sixth form is
P = X/N = 3/5 = 0.60
If we select all possible samples of the students who took
statistics in sixth form we have:

9. Statistics in Sixth Form

10. Example 5.3: Ms. Smarthead
ImaginethatMs.Smartheadisrunningfor
presidentoftheGuildofStudentsattheSt.
AugustinecampusofUWI.Sheclaimsthat
53%ofstudentsfavourherastheirchoice
forthepostofpresident.Assumethatthis
claimistrue.Whatistheprobabilitythatina
randomsampleof400studentsattheSt.
Augustinecampus,lessthan49%willfavour
herastheirchoiceforpresident?

11. ImaginethatMs.SmartheadisrunningforpresidentoftheGuildofStudentsat
theSt.AugustinecampusofUWI.Sheclaimsthat53%ofstudentsfavourheras
theirchoiceforthepostofpresident.Assumethatthisclaimistrue.Whatis
theprobabilitythatinarandomsampleof400studentsattheSt.Augustine
campus,lessthan49%willfavourherastheirchoiceforpresident?
Fromtheinformationgiven,weknowthatthepopulationproportion
p=0.53(53%)andthereforeq=0.47.
We shall use ෝ??????=0.49

12. ImaginethatMs.SmartheadisrunningforpresidentoftheGuildofStudentsat
theSt.AugustinecampusofUWI.Sheclaimsthat53%ofstudentsfavourheras
theirchoiceforthepostofpresident.Assumethatthisclaimistrue.Whatis
theprobabilitythatinarandomsampleof400studentsattheSt.Augustine
campus,lessthan49%willfavourherastheirchoiceforpresident?
Wearelookingfortheprobabilitythatlessthan49%willfavourheras
theirchoiceofpresident.
TheprobabilityweareinterestedinisfindingP(Ƹ??????<0.49).Therefore,
thecorrespondingzvalueisz=(0.49–0.53)/0.025=-1.60.
Hence,P(ෝ??????<0.49)=P(??????<-1.60)=0.0548.
Sothereisonlyasmallprobabilitythatlessthan49%willfavourher
aspresident.
Wecanconcludethattheprobabilitythatinarandomsampleof400
studentsattheSt.Augustinecampus,lessthan49%willfavourher
astheirchoiceforpresidentis0.0548.So,shehasaprettygood
chanceofbeingelectedpresident.

13. Looking at the problem another way…
ImaginethatMs.Smartheadisrunningfor
presidentoftheGuildofStudentsattheSt.
AugustinecampusofUWI.Sheclaimsthat80%of
studentsfavourherastheirchoiceforthepostof
president.Assumethatthisclaimistrue.Whatis
theprobabilitythatinarandomsampleof100
studentsattheSt.Augustinecampus,shewillbe
selectediftherequirementisatleastasimple
majorityinfavourwillconfirmherastheirchoice
forpresident?

14. Solution
ThisisanotherexampleofSamplingDistributionofSampleProportion.80%of
studentsfavourherastheirchoiceforthepostofpresidentsop=0.80andq=
0.20withn=100.
Stddev,σ
Ƹ
??????
=√(0.80*0.20)/100=0.04
Thez-valuefor(0.50-0.80)/0.04=-0.75
WearelookingforP(Ƹ??????>0.50)=P(z>-0.75)=77.34%soshehasagoodchance.

17. Questions