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 1st and 3rd examples above are simple.

• If there are several possible values of the parameter we call the hypothesis composite.– The 2nd and 4th examples above are composite.

Statistical Hypotheses

• Statistical hypothesis testing specifies two competing claims relating to the population. – One hypothesis is called the null hypothesis, H0

– The other is called the alternative hypothesis, Ha

• 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– a 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 unlikelyvalue under the null hypothesis

Statistical Hypotheses

• Summary table:

• a is often called the tests size and 1-β is called test’s power

• Sometimes a is called the false positive rate

• Sometimes b is called the false negative rate

H0 is really true H0 is really false

Test rejects H0 Type I error. Probability is α

OK

Test does not reject H0

OK Type II error. Probability is β

Statistical Hypotheses

Hypothesis Testing – Mean of Normal Distribution

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 1.518785. Given the sample of RIs below, test this hypothesis:

1.51881, 1.51874, 1.51883, 1.51865, 1.51878, 1.51882, 1.51876, 1.51877, 1.51882, 1.51882, 1.51883, 1.51882, 1.51882, 1.51882, 1.51879, 1.51876, 1.51886, 1.51879, 1.51877, 1.51880, 1.51881, 1.51881, 1.51887, 1.51877, 1.51883, 1.51872, 1.51883, 1.51875, 1.51880, 1.51877, 1.51881

One (large) sample hypothesis test that mean = 1.518785 vs. the alternative that the mean ≠ 1.518785

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 (1.518785) 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