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525201Statistics and Numerical Method
Part I: StatisticsWeek IV: Decision Making
(1 sample)
1/2555 สมศั�กดิ์�� ศั�วดิ์�รงพงศั�
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4-1 Statistical Inference
Sampling
Inference
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4-2 Point Estimation
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4-3 Hypothesis TestingStatistical hypothesis testing as
the data analysis stage of a comparative experiment, in which the engineer is interested
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Statistical HypothesisTwo side hypothesis
One side hypothesis
H0 : Null HypothesisH1 : Alternative Hypothesis
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Hypothesis TestingIf this information is consistent with the hypothesis, then we will conclude that the hypothesis is true;
If this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.
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Type I and II Errors
Ho TRUE
Fail to Reject Ho or
ACCEPT Ho
REJECT Ho
TYPE II ERROR
TYPE I ERRORProbability , aSignificance level
DecisionDecision
Probability , b
Ho FALSERealityReality
THE PROBABILITY OF TYPE I ERROR IS OFTEN SET AT 5%.THE PROBABILITY OF TYPE II ERROR IS OFTEN SET AT 10%
THE PROBABILITY OF TYPE I ERROR IS OFTEN SET AT 5%.THE PROBABILITY OF TYPE II ERROR IS OFTEN SET AT 10%
CONFIDENCE
Probability ,1 – aCorrect Decision
POWER
Probability, 1 – bCorrect Decision
Weak Conclusion
Strong Conclusion
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Decision criteria
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Type I error;
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Type II error;
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sample size and critical region
1.Reduce of critical region, alpha always increase2.Alpha and Beta are related, at remain sample size3.An increase in sample size will reduce both alpha
and beta4.When the null hypothesis is false, beta increase
as the true value of the parameter approaches the value hypothesized in the null hypothesis
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P-Value
P-Value is not the probability that the null hypothesis is false, nor is 1-P the probability that null hypothesis is true. The null hypothesis is either true or false, and so the proper interpretation of the P-value is in term of the risk of wrongly rejecting H0
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4-4 Inference on the mean of a population; variance known
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Reject H0 if the observed value of the test statistic z0 is either:
or
Fail to reject H0 if
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Ex: 0=50, =2, =0.05, n=25 and sample average=51.3H0: =50 and H1: ≠50=0.05 (two tails), -z0.025=-1.96
and z0.025=1.96z0=3.25P=2[1-(3.25)]=0.0012Since 0.0012<0.05 then reject H0Test of mu = 50 vs not = 50The assumed standard deviation = 2
N Mean SE Mean 95% CI Z P25 51.300 0.400 (50.516, 52.084) 3.25 0.001
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Ex: 0=50, =2, =0.05, n=16, 9, 4 and sample average=51.3
One-Sample Z
Test of mu = 50 vs not = 50The assumed standard deviation = 2
N Mean SE Mean 95% CI Z P16 51.300 0.500 (50.320, 52.280) 2.60
0.009
N Mean SE Mean 95% CI Z P9 51.300 0.667 (49.993, 52.607) 1.95 0.051
N Mean SE Mean 95% CI Z P4 51.30 1.00 (49.34, 53.26) 1.30 0.194
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Confidence interval on the mean
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Confidence interval on the mean
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4-5 Inference on the Mean of a Population, Variance Unknown
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Inference on the Mean of a Population, Variance Unknown
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Calculating P-value
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Ex: sampling n=15, mean=0.8375, s=0.02456, =0.05, test for exceed 0.82
H0: =0.82 and H1: >0.82=0.05 (upper tail), t0=2.72P=0.008Since 0.008<0.05 then reject H0
Test of mu = 0.82 vs > 0.82
95% LowerVariable N Mean StDev SE Mean Bound T PC1 15 0.83724 0.02456 0.00634 0.82607 2.72 0.008
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4-7 Inference on a Population Proportion (Binomial)
We will consider testing:
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Inference on a Population Proportion (Binomial)
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Ex: sampling 200 samples and found 4 defects. Please check the defect rate not exceed 0.05 with =0.05H0: p=0.05 and H1: p<0.05=0.05 (lower tail), z0=-1.95P=(-1.95)=0.0256Since 0.0256<0.05 then reject H0
Test of p = 0.05 vs p < 0.05
95% UpperSample X N Sample p Bound Z-Value P-Value1 4 200 0.020000 0.036283 -1.95 0.026
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Power and Sample Size1 sample, z-test2-side, power=, significant
level=, difference==-0
1-side, power=, significant level=, difference==-0
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1-Sample Z Test
Testing mean = null (versus not = null)Calculating power for mean = null + differenceAlpha = 0.05 Assumed standard deviation = 2
Sample TargetDifference Size Power Actual Power 1 43 0.9 0.906375
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Sample size and decision making
In general, if n 30, the sample variance s2 will be close to σ2 for most samples
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Power and Sample SizeSample and confident interval
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Power and Sample SizeT-testOperating Characteristic (OC
curves)
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Power and Sample Size1 proportion test