The diastolic blood pressure for American women aged 18-44 has approx The diastolic blood pressure for American women aged 18-44 has approx. a normal distribution with pop. Mean 75 mmHG and standard deviation 10 mmHg. We suspect that regular exercise will lower blood pressure. A random sample of 25 women who jog at least five miles a week gives sample mean blood pressure 71 mmHG. Is this good evidence that the mean diastolic blood pressure for the population of female regular exercisers is lower than 75 mmHG? A standard solution is supposed to have conductivity 5 microsiemens per centimeter. We know that measurements of conductivity aren’t perfectly precise: they vary according to a Normal distribution with mean equal to the true conductivity and standard deviation 0.2 microsiemens per centimeter. Six measurements of the solution’s conductivity are: 5.32 4.88 5.10 4.73 5.15 4.75 Is this evidence that the true conductivity (the mean of the population of all measurements) is not 5 microsiemens per centimeter?

11.3:Using significance tests Statistical significance is valued because it points to an effect that is unlikely to occur simply by chance Widely used in reporting the results of research in applied science, industry, and legal proceedings Some products require significant evidence of effectiveness and safety

Choosing a Level of Significance The purpose of a test of sig. is to give a clear statement of the degree of evidence provided by the sample against the null hypothesis = p-value!! Sometimes we will make a decision if our evidence reaches a certain standard, but we need a standard to set this against = level of significance Ex1: Drug companies use .01 level Ex2: Lawsuits alleging racial discrimination if the % hired of ethnic minorities hired is less than the .05 level

Choosing a Significance Level How plausible is the Null Hypothesis? If the Null represents an assumption that people have believed for years, your significance level should be small (need strong evidence) What are the consequences of rejecting the Null Hypothesis? If rejecting the Null means an expensive change, you need strong evidence that the change will bring about a profit or benefit to those having to bear the expense.

Significant/Insignificant There is no sharp border between “significant” and “Insignificant” – only increasingly strong evidence as the p-value decreases. Always better to report the p-value, which allows us to decide (individually) if the evidence is sufficiently strong. Statistical significance is not the same as practical importance! Pay attention to the actual data as well as the p-value. Plot your data!

Outliers and other considerations A few outlying observations can produce highly significant results if you blindly apply common tests of significance. Outliers can destroy the significance of otherwise convincing data. Faulty data collection and testing a hypothesis on the same data that suggested the hypothesis can invalidate a test. A confidence interval estimates the size of an effect rather than simply asking it its too large to reasonably occur by chance alone. P. 721, 11.43, 11.44 Statistical applet

11.4: Inference as Decision Link Calculators: Download “Power” program. Using significance tests with fixed alpha level points to the outcome of a test as a decision. If our result is significant at this level, we reject the null hypothesis in favor of the alternative hypothesis. Otherwise we fail to reject the null hypothesis. Tests of significance concentrate on the null hypothesis.

Scenario Present: Suspect and President of Student Court This student suspect has been arrested for stealing paper clips from the main office. The suspect claims, “I was only fiddling with the paper clips while waiting for an appointment with my counselor.” You, the class, are the student court. If the student is found guilty of the “crime,” the suspect will not be allowed to attend the next school dance. The president of the student court announces, “This student should be considered innocent unless there is sufficient evidence to find them guilty.” That is, the court’s hypothesis is that they are innocent, and it is looking to see if the evidence against them is sufficient to warrant rejecting that hypothesis and thus, finding them guilty. What type of errors can be made in this situation?

Type I/Type II Errors

Example 1 A producer of bearings and the consumer of the bearings agree that each carload must meet certain quality standards. When a carload arrives, the consumer inspects a sample of the bearings. On the basis of the sample outcome, the consumer makes some decision about whether or not to reject the carload.

Error Probabilities We assess any rule for making decisions by looking at the probabilities of the 2 types of error The mean diameter of a type of bearing is supposed to be 2.000 cm. The bearing diameters vary normally with standard deviation .010 cm. When a lot of the bearings arrives, the consumer takes an SRS of 5 bearings from the lot and measures their diameters. The consumer rejects the bearings if the sample mean diameter is significantly different from 2 at the 5% significance level. Find: P(Type I Error) P(Type II Error) when the mean is 2.015. Power

It is usual to report the probability that a test does reject the Null hypothesis when an alternative is true (= power) The higher this probability is, the more sensitive the test is High power is desirable! 80% power is becoming a standard In order to calculate power, fix an alpha so we have a fixed rule to reject Ho; usually .05. 4 ways to increase power: 1) Increase alpha 2) Consider an alternative far from your hypothesized value 3) Increase n 4) Decrease

P-value vs. Power P-value: Describes what happens if the null hypothesis is true = Assumes ho is true! Power: Describes what happens if the alternative hypothesis is true = Assumes ha is true! Decide what alternatives the test should detect and check that the power is adequate; power depends on what parameter for Ha we are interested in.

Many homeowners buy detectors to check for the invisible gas radon in their homes. We want to determine the accuracy of these detectors. To answer this question, university researchers placed 12 radon detectors in a chamber that exposed them to 105 picocuries per liter of radon. The detector readings were as follows: 91.9 97.8 111.4 122.3 105.4 95.0 103.8 99.6 96.6 119.3 104.8 101.7 Assume that = 9 picocuries per liter of radon for the population of all radon detectors. We want to determine if there is convincing evidence at the 10% significance level that the mean reading of all detectors of this type differs from the true value 105, so our hypotheses are H0: µ = 105 and Ha: µ 105. A significance test to answer this question was carried out. The test statistic is z = –0.3336, and the P-value is 0.74. 1) Describe what a Type I error would be in this situation. 2) Calculate the probability of a Type I error for this problem. 3) The researchers who carried out the study suspect that the large P-value is due to low power. First describe what a Type II error would be in this situation, then determine the probability of a Type II error when in fact µ = 100. Finally, compute the power of the test against the alternative. 4) If the sample size is increased to n = 30, what will be the power against the alternative, µ = 100? What happened to the power as the sample size increased?