Hypothesis Testing

“Not Guilty” In criminal proceedings in U.S. courts the defendant is presumed innocent until proven guilty and the prosecutor must prove the defendant guilty beyond a reasonable doubt. If the jury feels that the prosecutor has not adequately proven his case then they find the defendant not guilty. “not-guilty” is not necessarily the same as innocent. It just means not enough evidence to convict.

Statistical Hypotheses A statistical hypothesis is a claim about some characteristic or characteristics of a population. – Here are some hypothesis examples: The average (mean) height of female college students equals 63 inches. The average (mean) height of female college students is no more than 63 inches. The percentage of type A blood in the population of the United States is 40%. The percentage of type A blood in the population of the United States is not equal to 40% If the hypothesis completely specifies the parameter in question then it is called a simple hypothesis. – The 1 st and 3 rd examples above are simple. If there are several possible values of the parameter we call the hypothesis composite. – The 2 nd and 4 th examples above are composite.

Statistical Hypotheses Statistical hypothesis testing specifies two competing claims relating to the population. – One hypothesis is called the null hypothesis, H 0 – The other is called the alternative hypothesis, H a A test of hypothesis is a procedure using sample data to test whether the null hypothesis should be rejected or not. – The null hypothesis is not rejected unless the sample data (i.e. the “evidence”) “strongly” indicates that the null hypothesis is “unlikely”.

Statistical Hypotheses Choose a probability of rejecting the null hypothesis when it is actually true –  level of significance, usually takes on the value of.05 or.01. Why?? Fisher said so… If the test rejects a true null hypothesis then we have made what is called a type I error. – The probability of type I error = α. – Making a type I error is like convicting an innocent defendant. If the test fails to reject the null hypothesis when it is actually false we made a type II error. – Probability of a type II error is denoted by β.

Assume/derive a “null” probability model for a statistic E.g. sample averages follow a Gaussian curve Say sample statistic falls here “Wow”! That’s an unlikely value under the null hypothesis Statistical Hypotheses

Summary table:  is often called the tests size and 1-β  is called test’s power Sometimes  is called the false positive rate Sometimes  is called the false negative rate H 0 is really trueH 0 is really false Test rejects H 0 Type I error. Probability is α OK Test does not reject H 0 OKType II error. Probability is β Statistical Hypotheses

Hypothesis Testing – Mean of Normal Distribution

and/or if sample is large, n>30 and needs to be estimated from a small sample

Hypothesis Testing – Mean of Normal Distribution

Hypothesis Testing – Proportion-1

Hypothesis Testing – Mean of Normal Distribution

Hypothesis Testing – Example Assume the mean RI of a pre-annealed glass pane is Given the sample of RIs below, test this hypothesis: , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , One (large) sample hypothesis test that mean = vs. the alternative that the mean ≠

How would this calculation be different if we did the one- sided hypothesis?

Hypothesis Testing – Example Follow-up: What is the 95% confidence interval for the mean given this sample. Where does the assume population mean ( ) fall with respect to this CI? Given your finding, does it make sense that we could not reject the null? 95% confidence interval covers the stated population mean