INFERENTIAL STATISTICS ESTIMATION POINT ESTIMATION HYPOTHESIS TESTING INTERVAL

INFERENTIAL STATISTICS ESTIMATION POINT ESTIMATION HYPOTHESIS TESTING INTERVAL ESTIMATION

POINT ESTIMATE OF PROCESS PARAMETERS • A random variable is characterized or described by its probability distribution. This distribution is described by its parameters. • We may define an estimator of an unknown parameter as a statistics that corresponds to a paramater. • A particular numerical value of an estimator, computed form a sample data is called an estimate

• Sample standard deviation is not an unbiased estimator of the population SD.

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

• H 1 in equation 3 -22 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 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 on Mean, Variance Known Furthermore, a 100(1 − α)% upper confidence bound on µ is whereas a 100(1 − α)% lower confidence bound on µ is

• 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

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