Statistics Chapter 10 Section 4

Jerzy Nehman 1934 – introduced confidence intervals 1894 – 1981 Trained in Poland Worked in agriculture Moved to London in 1934 1938 – Cal Berkeley Very well published

Inference as decision Tests of significance assess the strength of evidence against the null hypothesis A level of significance chosen in advance indicates a decision.

Acceptance sampling Popularized by Dodge and Romig Applied to U.S. military testing in WWII Testing of bullets Originally called Lot Acceptance Sampling

Example – potato chips Do you test the entire batch or just some of them?

Hypothesis H0: the batch of potato chips meets standards Ha: the potato chips do not meet standards

BUT What if we reject the batch but should have kept them, or we accepted (yep I said accepted) the batch but should have discarded them? THESE ARE ERRORS in DECISION

Type I and Type II Errors If we reject Ho (accept Ha) when in fact H0 is true, this is a Type I error If we accept (reject Ha) H0 when in fact Ha is true, this is a Type II error.

Example from Wikipedia consider the case where a patient is being tested for HIV. Typically, the null hypothesis is that he or she does not have the disease, while the alternative hypothesis is that HIV is present. If the null hypothesis is rejected when it is in fact true (and the patient is well), this is a Type I error. In this example, because the test’s results suggest illness (i.e., do not reject the alternative hypothesis of illness), it is also known as a “false positive.” A Type II error occurs when a null hypothesis is not rejected despite being false. In this case, a “false negative” gives the patient a false illusion of health

False positive – don’t have it but treated anyway False negative – have it but told you don’t

Significance and Type I error The significance level of any fixed level test is the PROBABILITY of a Type I error. So when we ask for the probability of a type I error all it is, is the significance level for the test.

Calculation of a Type II error

Step 1 Write the rule for accepting H0 in terms of x.

Step 2 Find the probability of accepting H0 assuming that the alternative is true.

Probability of a Type II error Probability of a Type II error. This is the probability that the test accepts H0 when the alternative hypothesis is true.

Power The probability that a fixed level α significance test will reject H0 when a particular alternative value of the parameter is true is called the power of the test against that alternative. The power of a test against any alternative is 1 minus the probability of a Type II error for that alternative.

So what Calculations of P values and power say What would happen if we repeated the test many times P-value describes what would happen supposing that the null hypothesis is true. Power describes what would happen supposing that a particular alternative is true.

How to answer the question of power State H0 and Ha, the particular alternative we want to detect, and the significance level α Find the values of x that will lead us to reject H0. Calculate the probability of observing these values of x when the alternative is true.

Step 1

Step 2

Step 3

Increasing the power Suppose you find that the power is too small. – What can you do to increase it? Increase α Consider a Ha that is farther away from µ0 Increase sample size Decrease σ