Chapter 4 Inference about Process Quality Random Sample

Chapter 4. Inference about Process Quality

Random Sample Statistics

Chi-square ( 2) Distribution

t Distribution

F Distribution

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

• As n gets large the bias goes to zero

Hypothesis Testing Alternative Hypothesis Null Hypothesis • In this example, H 1 is a two-sided alternative hypothesis

• H 1 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 H 0 if |Z 0| > Z /2, where Z /2 is the upper /2 percentage point of the standard normal distribution.

One-Sided Alternative Hypotheses • In some situations we may wish to reject H 0 only if the true mean is larger than µ 0 – Thus, the one-sided alternative hypothesis is H 1: µ>µ 0, and we would reject H 0: µ=µ 0 only if Z 0>Zα • If rejection is desired only when µ<µ 0 – Then the alternative hypothesis is H 1: µ<µ 0, and we reject H 0 only if Z 0<−Zα

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

• For the two-sided alternative hypothesis, reject H 0 if |t 0| > 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 H 1: µ 1 > µ 0, reject H 0 if t 0 > tα, n − 1, and • If H 1: µ 1 < µ 0, reject H 0 if t 0 < −tα, n − 1 • One could also compute the P-value for a t-test

t 0. 025, 14 = 2. 145. Thus, we should accept H 0.

• Section 3 -3. 4 describes hypothesis testing and confidence intervals on the variance of a normal distribution

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

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.