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Confidence Interval Estimation()

*Confidence Intervals (Population Mean, ) Population Standard Deviation is Known Population Standard Deviation is Unknown (Population Proportion, p) (Sample Size, n)

*Point and Interval Estimatespoint estimate confidence interval Point EstimateLower Confidence LimitUpperConfidence LimitWidth of confidence interval

*Population Parameter

Point Estimates(a Point Estimate)MeanProportionpspX

*General FormulaPoint Estimate (Critical Value)(Standard Error)

Confidence Level, (1-) confidence level = 95% (1 - ) = .95:, 95% 5%(continued)

Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion Known

Confidence Interval for ( Known) Assumptions

Confidence interval :

(Z (critical value) /2)

-*, ZConsider a 95% confidence interval:Z= -1.96Z= 1.96Point EstimateLower Confidence LimitUpperConfidence LimitZ units:X units:Point Estimate0

Example 11 2.20 0.35 .

95% Solution:(continued)

Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion Known

standard deviation unknown, sample standard deviation, S extra uncertainty S sample to sample the t distribution the normal distributionConfidence Interval for ( Unknown)

(Assumptions)Population standard deviation is unknownPopulation is normally distributed population normal Students t Distribution :

( t critical value t distribution df = n-1 /2 Curve) Confidence Interval for ( Unknown) (continued)

If the mean of these three values is 8.0, then X3 must be 9 (X3 )Degrees of Freedom (df), n = 3, degrees of freedom = n 1 = 3 1 = 2Idea: Number of observations that are free to vary after sample mean has been calculatedExample: 3 8

Let X1 = 7Let X2 = 8What is X3?

Students t Distributiont0t (df = 5) t (df = 13)t-distributions bell-shaped symmetric, normalStandard Normal(t with df = )Note: t Z as n increases

Students t TableUpper Tail Areadf

.25.10.0511.0003.0786.31420.8171.8862.92030.7651.6382.353t02.920 t values, probabilitiesLet: n = 3 df = n - 1 = 2 = .10 /2 =.05/2 = .05

Example n = 25 , = 50 S = 8. 95% confidence interval

df. = n 1 = 24,

The confidence interval (46.698 , 53.302)

Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion Known

Confidence Intervals for the Population Proportion, p sample proportion normal distribution sample size , standard deviation

sample data :

Confidence Interval Endpoints

Z standard normal value for the level of confidence desiredps sample proportionn sample size

Example 100 25 95% confidence interval

(Sampling Error) sampling error (e) (1 - )

For the MeanDeterminingSample SizeSampling error (margin of error)

Sample SizeFor the MeanDeterminingSample Size(continued) n

= 45, 5 90% confidence level sample standard deviation, S

Sample Size DeterminingSample SizeFor theProportion

Determining Sample SizeDeterminingSample SizeFor theProportion n (continued)

Determining Sample Size

level of confidence (1 - ), critical Z value sampling error, e proportion successes, pp (continued)

proportion 3%, 95% confidence level ( ps = .12)Solution:For 95% confidence, use Z = 1.96e = .03ps = .12, so use this to estimate p

* 1000 1000 80 87.6 . 22.3 . 100000 . 95% 1000 100000 .

*Solution