Sample size for an interval estimate of a population mean. Interval estimation of a population...

•Sample size for an interval estimate of a population mean.

•Interval estimation of a population proportion.

•Sample size for an interval estimate of a population proportion.

Let Let EE = the maximum sampling error mentioned in = the maximum sampling error mentioned in the precision statement.the precision statement. Let Let EE = the maximum sampling error mentioned in = the maximum sampling error mentioned in the precision statement.the precision statement.

EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate. EE is the amount added to and subtracted from the is the amount added to and subtracted from the point estimate to obtain an interval estimate.point estimate to obtain an interval estimate.

Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Meanof a Population Mean

EE is often referred to as the is often referred to as the margin of error.margin of error. EE is often referred to as the is often referred to as the margin of error.margin of error.


Margin of ErrorMargin of Error

Necessary Sample Size Necessary Sample Size

E zn

/2E zn

/2

nz

E

( )/ 22 2

2n

z

E

( )/ 22 2

2

Recall that Discount Sounds is evaluating a potential location for a new retail outlet, based in part, on the mean annual income of the individuals inthe marketing area of the new location.

Suppose that Discount Sounds’ management team wants an estimate of the population mean such that there is a .95 probability that the sampling error is $500 or less.

How large a sample size is needed to meet therequired precision?

SSDD


At 95% confidence, z.025 = 1.96. Recall that = 4,500.

zn

/2 500zn

/2 500

2 2

2

(1.96) (4,500)311.17 312

(500)n

2 2

2

(1.96) (4,500)311.17 312

(500)n


SSDD

A sample of size 312 is needed to reach a desiredA sample of size 312 is needed to reach a desired

precision of precision of ++ $500 at 95% confidence. $500 at 95% confidence.

/2/2 /2/2

Interval EstimationInterval Estimationof a Population Proportionof a Population Proportion

Normal Approximation of Sampling Distribution Normal Approximation of Sampling Distribution of When of When npnp >> 5 and 5 and nn(1 – (1 – pp) ) >> 5 5pp

Samplingdistribution of

Samplingdistribution of pp

(1 )p

p p

n

(1 )p

p p

n

pppp

/ 2 pz / 2 pz / 2 pz / 2 pz

1 - of all values1 - of all valuespp

p

p

• Interval Estimate

Interval Estimationof a Population Proportion

p zp pn

/

( )2

1p z

p pn

/

( )2

1

where: 1 -where: 1 - is the confidence coefficient is the confidence coefficient

zz/2 /2 is the is the zz value providing an area of value providing an area of

/2 in the upper tail of the standard/2 in the upper tail of the standard

normal probability distributionnormal probability distribution

is the sample proportionis the sample proportionpp

Example: Political Science, Inc.

• Interval Estimation of a Population Proportion

Political Science, Inc. (PSI)specializes in voter polls andsurveys designed to keeppolitical office seekers informedof their position in a race.

Using telephone surveys, PSI interviewers askregistered voters who they would vote for if theelection were held that day.

Interval Estimation of a Population Proportion Interval Estimation of a Population Proportion

In a current election campaign, In a current election campaign, PSI has just found that 220PSI has just found that 220registered voters, out of 500registered voters, out of 500contacted, favor a particularcontacted, favor a particular

candidate.candidate.PSI wants to develop a 95% confidence PSI wants to develop a 95% confidence

interval interval estimate for the proportion of the population ofestimate for the proportion of the population ofregistered voters that favor the candidate.registered voters that favor the candidate.

Example: Political Science, Inc.Example: Political Science, Inc.

p zp pn

/

( )2

1p z

p pn

/

( )2

1

where: where: nn = 500, = 220/500 = .44, = 500, = 220/500 = .44, zz/2 /2 = = 1.961.96pp

Interval EstimationInterval Estimationof a Population Proportionof a Population Proportion

PSI is 95% confident that the proportion of all votersPSI is 95% confident that the proportion of all voters

that favor the candidate is between .3965 and .4835.that favor the candidate is between .3965 and .4835.

.44(1 .44).44 1.96

500

.44(1 .44)

.44 1.96500

= .44 + .0435= .44 + .0435

Using Excel to ConstructUsing Excel to Constructa Confidence Intervala Confidence Interval

Formula WorksheetFormula WorksheetA B C

1 Favor Sample Size 5002 Yes Total Yes =COUNTIF(A2:A501,"Yes")3 Yes Sample Proportion =C2/C14 No 5 Yes Confid. Coefficient 0.956 No Lev. of Signif. (alpha) =1-C57 No z Value =NORMSINV(1-C6/2)8 No9 No Standard Error =SQRT(C3*(1-C3)/C1)10 Yes Margin of Error =C7*C911 No12 Yes Point Estimate =C313 No Lower Limit =C12-C1014 No Upper Limit =C12+C10

Note: Rows 15-501 are not shown.Note: Rows 15-501 are not shown.

Value WorksheetValue Worksheet

Using Excel to ConstructUsing Excel to Constructa Confidence Intervala Confidence Interval

A B C

1 Favor Sample Size 5002 Yes Total Yes 2203 Yes Sample Proportion 0.44004 No 5 Yes Confid. Coefficient 0.956 No Lev. of Signif. (alpha) 0.057 No z Value 1.96008 No9 No Standard Error 0.0222010 Yes Margin of Error 0.0435111 No12 Yes Point Estimate 0.440013 No Lower Limit 0.396514 No Upper Limit 0.4835

Note: Rows 15-501 are not shown.Note: Rows 15-501 are not shown.

Margin of ErrorMargin of Error

Necessary Sample SizeNecessary Sample Size

Sample Size for an Interval EstimateSample Size for an Interval Estimateof a Population Proportionof a Population Proportion

E zp pn

/

( )2

1E z

p pn

/

( )2

1

nz p p

E

( ) ( )/ 22

2

1n

z p p

E

( ) ( )/ 22

2

1

Suppose that PSI would like a .99 probabilitythat the sample proportion is within + .03 of the population proportion.

How large a sample size is needed to meet the required precision?

Sample Size for an Interval Estimateof a Population Proportion

At 99% confidence, At 99% confidence, zz.005.005 = 2.576. Recall that = 2.576. Recall that = .44.= .44.

pp

A sample of size 1817 is needed to reach a desiredA sample of size 1817 is needed to reach a desired

precision of precision of ++ .03 at 99% confidence. .03 at 99% confidence.

2 2/ 2

2 2

( ) (1 ) (2.576) (.44)(.56) 1817

(.03)

z p pn

E

2 2

/ 2

2 2

( ) (1 ) (2.576) (.44)(.56) 1817

(.03)

z p pn

E


/ 2

(1 ).03

p pz

n

/ 2

(1 ).03

p pz

n

Note:Note: We used .44 as the best estimate of We used .44 as the best estimate of pp in in thethepreceding expression. If no information is preceding expression. If no information is availableavailableabout about pp, then .5 is often assumed because it , then .5 is often assumed because it providesprovidesthe highest possible sample size. If we had the highest possible sample size. If we had usedusedpp = .5, the recommended = .5, the recommended nn would have been would have been 1843.1843.


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