15.copnfidence interval print
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Confidence Interval Estimation()
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*Confidence Intervals (Population Mean, ) Population Standard Deviation is Known Population Standard Deviation is Unknown (Population Proportion, p) (Sample Size, n)
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*Point and Interval Estimatespoint estimate confidence interval Point EstimateLower Confidence LimitUpperConfidence LimitWidth of confidence interval
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*Population Parameter
Point Estimates(a Point Estimate)MeanProportionpspX
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*General FormulaPoint Estimate (Critical Value)(Standard Error)
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Confidence Level, (1-) confidence level = 95% (1 - ) = .95:, 95% 5%(continued)
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Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion Known
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Confidence Interval for ( Known) Assumptions
Confidence interval :
(Z (critical value) /2)
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-*, ZConsider a 95% confidence interval:Z= -1.96Z= 1.96Point EstimateLower Confidence LimitUpperConfidence LimitZ units:X units:Point Estimate0
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Example 11 2.20 0.35 .
95% Solution:(continued)
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Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion Known
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standard deviation unknown, sample standard deviation, S extra uncertainty S sample to sample the t distribution the normal distributionConfidence Interval for ( Unknown)
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(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)
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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?
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Students t Distributiont0t (df = 5) t (df = 13)t-distributions bell-shaped symmetric, normalStandard Normal(t with df = )Note: t Z as n increases
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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
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Example n = 25 , = 50 S = 8. 95% confidence interval
df. = n 1 = 24,
The confidence interval (46.698 , 53.302)
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Confidence IntervalsPopulation Mean UnknownConfidenceIntervalsPopulationProportion Known
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Confidence Intervals for the Population Proportion, p sample proportion normal distribution sample size , standard deviation
sample data :
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Confidence Interval Endpoints
Z standard normal value for the level of confidence desiredps sample proportionn sample size
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Example 100 25 95% confidence interval
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(Sampling Error) sampling error (e) (1 - )
For the MeanDeterminingSample SizeSampling error (margin of error)
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Sample SizeFor the MeanDeterminingSample Size(continued) n
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= 45, 5 90% confidence level sample standard deviation, S
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Sample Size DeterminingSample SizeFor theProportion
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Determining Sample SizeDeterminingSample SizeFor theProportion n (continued)
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Determining Sample Size
level of confidence (1 - ), critical Z value sampling error, e proportion successes, pp (continued)
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proportion 3%, 95% confidence level ( ps = .12)Solution:For 95% confidence, use Z = 1.96e = .03ps = .12, so use this to estimate p
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* 1000 1000 80 87.6 . 22.3 . 100000 . 95% 1000 100000 .
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*Solution
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