1. Chapter 4. Inference about Process Quality

2. Random Sample
Statistics

3. Chi-square (2) Distribution

5. t Distribution

7. F Distribution

10. Estimator: estimates probability parameter from samples
Good Characteristics for Estimators
Unbiased
Minimum variance

11. As n gets large the bias goes to zero

13. Alternative Hypothesis
Null Hypothesis
Alternative Hypothesis
In this example, H1 is a two-sided alternative hypothesis
Hypothesis Testing

14. H1 is a two-sided alternative hypothesis.
The procedure for testing this hypothesis is to:
take a random sample of n observations on the random variable x,
compute the test statistic, and
reject H0 if |Z0| > Z/2, where Z/2 is the upper /2 percentage point of the standard normal distribution.

15. One-Sided Alternative Hypotheses
In some situations we may wish to reject H0 only if the true mean is larger than µ0
Thus, the one-sided alternative hypothesis is H1: µ>µ0, and we would reject H0: µ=µ0 only if Z0>Zα
If rejection is desired only when µ<µ0
Then the alternative hypothesis is H1: µ<µ0, and we reject H0 only if Z0<−Zα

17. Confidence Interval → If P ( L ≤ μ ≤ U ) = 1- α
L ≤ μ ≤ U is 100 (1- α) % confidence interval. →
If the variance is known.

20. For the two-sided alternative hypothesis, reject H0 if |t0| > t/2,n-1, where t/2,n-1, is the upper /2 percentage of the t distribution with n  1 degrees of freedom
For the one-sided alternative hypotheses,
If H1: µ1 > µ0, reject H0 if t0 > tα,n − 1, and
If H1: µ1 < µ0, reject H0 if t0 < −tα,n − 1
One could also compute the P-value for a t-test

22. t0.025, 14 = 2.145. Thus, we should accept H0.

23. Section describes hypothesis testing and confidence intervals on the variance of a normal distribution

24. Suppose, out of n samples chosen, x samples belongs to a subclass with probability p.

26. Confidence Intervals on a Population Proportion
For large n and p, use normal approximation.
For large n and small p, use Poisson approximation.
For small n, use binomial distribution.

46. Two independent samples of size n1 and n2.
Of them, x1 and x2 belong to the class of interest.

49. More Two Populations

50. Analysis of Variance (ANOVA)

51. If H0 is true:
If H1 is true:

52. For hypothesis H0 testing, use
with a-1 and a(n-1) degrees of freedom.
Alternative formulas for computing efficiency