1. 1/2555 สมศักดิ์ ศิวดำรงพงศ์ somsaksi@sut.ac.th
Statistics and Numerical Method Part I: Statistics Week IV: Decision Making (1 sample)
1/2555
สมศักดิ์ ศิวดำรงพงศ์

2. 4-1 Statistical Inference
Sampling
Inference

3. 4-2 Point Estimation

4. 4-3 Hypothesis Testing
Statistical hypothesis testing as the data analysis stage of a comparative experiment, in which the engineer is interested

5. Statistical Hypothesis
Two side hypothesis
One side hypothesis
H0 : Null Hypothesis
H1 : Alternative Hypothesis

6. Hypothesis Testing
If 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.

7. Type I and II Errors Reality Decision Weak Conclusion
Ho TRUE
Fail to Reject Ho or
ACCEPT Ho
REJECT Ho
TYPE II ERROR
TYPE I ERROR
Probability , a
Significance level
Decision
Probability , b
Ho FALSE
Reality
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 – a
Correct Decision
POWER
Probability, 1 – b
Weak Conclusion
Strong Conclusion

8. Decision criteria

9. Type I error; 

10. Type II error; 

11. sample size and critical region
Reduce of critical region, alpha always increase
Alpha and Beta are related, at remain sample size
An increase in sample size will reduce both alpha and beta
When the null hypothesis is false, beta increase as the true value of the parameter approaches the value hypothesized in the null hypothesis

12. 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

13. 4-4 Inference on the mean of a population; variance known

14. Reject H0 if the observed value of the test statistic z0 is either:or
Fail to reject H0 if

16. Ex: 0=50, =2, =0.05, n=25 and sample average=51.3
H0: =50 and H1: ≠50
=0.05 (two tails), -z0.025=-1.96 and z0.025=1.96
z0=3.25
P=2[1-(3.25)]=0.0012
Since <0.05 then reject H0
Test of mu = 50 vs not = 50
The assumed standard deviation = 2
N Mean SE Mean % CI Z P
(50.516, )

17. Ex: 0=50, =2, =0.05, n=16, 9, 4 and sample average=51.3
One-Sample Z
Test of mu = 50 vs not = 50
The assumed standard deviation = 2
N Mean SE Mean % CI Z P
(50.320, )
(49.993, )
N Mean SE Mean % CI Z P
(49.34, 53.26)

18. Confidence interval on the mean

19. Confidence interval on the mean

20. 4-5 Inference on the Mean of a Population, Variance Unknown

21. Inference on the Mean of a Population, Variance Unknown

22. Calculating P-value

23. Ex: sampling n=15, mean=0. 8375, s=0. 02456, =0
Ex: sampling n=15, mean=0.8375, s= , =0.05, test for  exceed 0.82
H0: =0.82 and H1: >0.82
=0.05 (upper tail), t0=2.72
P=0.008
Since 0.008<0.05 then reject H0
Test of mu = 0.82 vs > 0.82
95% Lower
Variable N Mean StDev SE Mean Bound T P C

24. 4-7 Inference on a Population Proportion (Binomial)
We will consider testing:

25. Inference on a Population Proportion (Binomial)

26. Ex: sampling 200 samples and found 4 defects
Ex: sampling 200 samples and found 4 defects. Please check the defect rate not exceed 0.05 with =0.05
H0: p=0.05 and H1: p<0.05
=0.05 (lower tail), z0=-1.95
P=(-1.95)=0.0256
Since <0.05 then reject H0
Test of p = 0.05 vs p < 0.05
95% Upper
Sample X N Sample p Bound Z-Value P-Value

27. Power and Sample Size 1 sample, z-test
2-side, power=, significant level=, difference==-0
1-side, power=, significant level=, difference==-0

28. 1-Sample Z Test
Testing mean = null (versus not = null)
Calculating power for mean = null + difference
Alpha = Assumed standard deviation = 2
Sample Target
Difference Size Power Actual Power

29. Sample size and decision making
In general, if n  30, the sample variance s2 will be close to σ2 for most samples

30. Power and Sample Size
Sample and confident interval

31. Power and Sample Size
T-test
Operating Characteristic (OC curves)

32. Power and Sample Size
1 proportion test