1)1) The diastolic blood pressure for American women aged The diastolic blood pressure for American women aged 18-44 has approx. a normal distribution with pop. Mean 75 18-44 has approx. a normal distribution with pop. Mean 75 mmHG and standard deviation 10 mmHg. We suspect that mmHG and standard deviation 10 mmHg. We suspect that regular exercise will lower blood pressure. A random regular exercise will lower blood pressure. A random sample of 25 women who jog at least five miles a week sample of 25 women who jog at least five miles a week gives sample mean blood pressure 71 mmHG. Is this good gives sample mean blood pressure 71 mmHG. Is this good evidence that the mean diastolic blood pressure for the evidence that the mean diastolic blood pressure for the population of female regular exercisers is lower than 75 population of female regular exercisers is lower than 75 mmHG?mmHG?

2)2) A standard solution is supposed to have conductivity 5 A standard solution is supposed to have conductivity 5 microsiemens per centimeter. We know that microsiemens per centimeter. We know that measurements of conductivity aren’t perfectly precise: they measurements of conductivity aren’t perfectly precise: they vary according to a Normal distribution with mean equal to vary according to a Normal distribution with mean equal to the true conductivity and standard deviation 0.2 the true conductivity and standard deviation 0.2 microsiemens per centimeter. Six measurements of the microsiemens per centimeter. Six measurements of the solution’s conductivity are:solution’s conductivity are:5.325.32 4.884.88 5.105.10 4.734.73 5.155.15 4.754.75Is this evidence that the true conductivity (the mean of the Is this evidence that the true conductivity (the mean of the population of all measurements) is not 5 microsiemens per population of all measurements) is not 5 microsiemens per centimeter?centimeter?

11.3:Using significance tests11.3:Using significance tests

Statistical significance is valued because it Statistical significance is valued because it points to an effect that is unlikely to occur points to an effect that is unlikely to occur simply by chancesimply by chance

Widely used in reporting the results of Widely used in reporting the results of research in applied science, industry, and research in applied science, industry, and legal proceedingslegal proceedings

Some products require significant Some products require significant evidence of effectiveness and safetyevidence of effectiveness and safety

Choosing a Level of SignificanceChoosing a Level of Significance

The purpose of a test of sig. is to give a clear The purpose of a test of sig. is to give a clear statement of the degree of evidence provided by statement of the degree of evidence provided by the sample against the null hypothesis = p-the sample against the null hypothesis = p-value!!value!!

Sometimes we will make a decision if our Sometimes we will make a decision if our evidence reaches a certain standard, but we evidence reaches a certain standard, but we need a standard to set this against = level of need a standard to set this against = level of significancesignificance

Ex1: Drug companies use .01 levelEx1: Drug companies use .01 level Ex2: Lawsuits alleging racial discrimination if the Ex2: Lawsuits alleging racial discrimination if the

% hired of ethnic minorities hired is less than the % hired of ethnic minorities hired is less than the .05 level.05 level

Choosing a Significance LevelChoosing a Significance Level

How plausible is the Null Hypothesis?How plausible is the Null Hypothesis?If the Null represents an assumption that people If the Null represents an assumption that people have believed for years, your significance level have believed for years, your significance level should be small (need strong evidence)should be small (need strong evidence)

What are the consequences of rejecting the Null What are the consequences of rejecting the Null Hypothesis?Hypothesis?If rejecting the Null means an expensive change, If rejecting the Null means an expensive change, you need strong evidence that the change will you need strong evidence that the change will bring about a profit or benefit to those having to bring about a profit or benefit to those having to bear the expense.bear the expense.

Significant/InsignificantSignificant/Insignificant

There is no sharp border between “significant” There is no sharp border between “significant” and “Insignificant” – only increasingly strong and “Insignificant” – only increasingly strong evidence as the p-value decreases.evidence as the p-value decreases.

Always better to report the p-value, which allows Always better to report the p-value, which allows us to decide (individually) if the evidence is us to decide (individually) if the evidence is sufficiently strong.sufficiently strong.

Statistical significanceStatistical significance is not the same as is not the same as practical importancepractical importance! Pay attention to the actual ! Pay attention to the actual data as well as the p-value. Plot your data!data as well as the p-value. Plot your data!

Outliers and other considerationsOutliers and other considerations

A few outlying observations can produce highly A few outlying observations can produce highly significant results if you blindly apply common tests significant results if you blindly apply common tests of significance.of significance.

Outliers can destroy the significance of otherwise Outliers can destroy the significance of otherwise convincing data.convincing data.

Faulty data collection and testing a hypothesis on the Faulty data collection and testing a hypothesis on the same data that suggested the hypothesis can same data that suggested the hypothesis can invalidate a test.invalidate a test.

A confidence interval estimates the size of an effect A confidence interval estimates the size of an effect rather than simply asking it its too large to reasonably rather than simply asking it its too large to reasonably occur by chance alone. occur by chance alone.

P. 721, 11.43, 11.44 Statistical appletP. 721, 11.43, 11.44 Statistical applet

11.4: Inference as Decision11.4: Inference as Decision

Link Calculators: Download “Power” program.Link Calculators: Download “Power” program. Using significance tests with fixed alpha level Using significance tests with fixed alpha level

points to the outcome of a test as a points to the outcome of a test as a decision. decision. If our result is significant at this level, we reject If our result is significant at this level, we reject

the null hypothesis in favor of the alternative the null hypothesis in favor of the alternative hypothesis. Otherwise we hypothesis. Otherwise we fail to reject fail to reject the null the null hypothesis.hypothesis.

Tests of significance concentrate on the null Tests of significance concentrate on the null hypothesis.hypothesis.

ScenarioScenario Present: Suspect and President of Student CourtPresent: Suspect and President of Student Court This student suspect has been arrested for stealing paper This student suspect has been arrested for stealing paper

clips from the main office. clips from the main office. The suspectThe suspect claims, “I was only fiddling with the paper claims, “I was only fiddling with the paper

clips while waiting for an appointment with my counselor.”clips while waiting for an appointment with my counselor.” You, the class, are the student court. If the student is You, the class, are the student court. If the student is

found guilty of the “crime,” the suspect will not be allowed found guilty of the “crime,” the suspect will not be allowed to attend the next school dance.to attend the next school dance.

The presidentThe president of the student court announces, “This of the student court announces, “This student should be considered innocent unless there is student should be considered innocent unless there is sufficient evidence to find them guilty.”sufficient evidence to find them guilty.”

That is, the court’s hypothesis is that they are innocent, That is, the court’s hypothesis is that they are innocent, and it is looking to see if the evidence against them is and it is looking to see if the evidence against them is sufficient to warrant rejecting that hypothesis and thus, sufficient to warrant rejecting that hypothesis and thus, finding them guilty.finding them guilty.

What type of errors can be made in this situation?What type of errors can be made in this situation?

Type I/Type II ErrorsType I/Type II Errors

Example 1Example 1

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

Error ProbabilitiesError ProbabilitiesWe assess any rule for making decisions by looking at the probabilities We assess any rule for making decisions by looking at the probabilities

of the 2 types of errorof the 2 types of error

The mean diameter of a type of bearing is supposed to The mean diameter of a type of bearing is supposed to be 2.000 cm. The bearing diameters vary normally with be 2.000 cm. The bearing diameters vary normally with standard deviation .010 cm. When a lot of the bearings standard deviation .010 cm. When a lot of the bearings arrives, the consumer takes an SRS of 5 bearings from arrives, the consumer takes an SRS of 5 bearings from the lot and measures their diameters. The consumer the lot and measures their diameters. The consumer rejects the bearings if the sample mean diameter is rejects the bearings if the sample mean diameter is significantly different from 2 at the 5% significance significantly different from 2 at the 5% significance level. Find:level. Find:

P(Type I Error)P(Type I Error) P(Type II Error) when the mean is 2.015.P(Type II Error) when the mean is 2.015. PowerPower

It is usual to report the probability that a test It is usual to report the probability that a test does does reject the reject the Null hypothesis when an alternative is true (= power)Null hypothesis when an alternative is true (= power)

The higher this probability is, the more sensitive the test isThe higher this probability is, the more sensitive the test is High power is desirable!High power is desirable! 80% power is becoming a standard80% power is becoming a standard In order to calculate power, fix an alpha so we have a fixed In order to calculate power, fix an alpha so we have a fixed

rule to reject Ho; usually .05.rule to reject Ho; usually .05. 4 ways to increase power4 ways to increase power::

1) Increase alpha1) Increase alpha2) Consider an alternative far from your hypothesized value2) Consider an alternative far from your hypothesized value3) Increase n3) Increase n4) Decrease 4) Decrease

P-value vs. PowerP-value vs. Power

P-valueP-value: Describes what happens if the null : Describes what happens if the null hypothesis is true hypothesis is true

= Assumes ho is true!= Assumes ho is true! PowerPower: Describes what happens if the : Describes what happens if the

alternative hypothesis is true alternative hypothesis is true = Assumes ha is true!= Assumes ha is true!

Decide what alternatives the test should detect Decide what alternatives the test should detect and check that the power is adequate; power and check that the power is adequate; power depends on what parameter for Ha we are depends on what parameter for Ha we are interested in.interested in.

Many homeowners buy detectors to check for the invisible gas radon in their 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 homes. We want to determine the accuracy of these detectors. To answer this question, university researchers placed 12 radon detectors in a this question, university researchers placed 12 radon detectors in a chamber that exposed them to 105 picocuries per liter of radon. The chamber that exposed them to 105 picocuries per liter of radon. The detector readings were as follows:detector readings were as follows:

91.9 97.8 111.4 122.3 105.4 95.0 103.8 91.9 97.8 111.4 122.3 105.4 95.0 103.8 99.6 96.6 119.3 104.8 101.7 99.6 96.6 119.3 104.8 101.7

Assume that = 9 picocuries per liter of radon for the population of all radon 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 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 10% significance level that the mean reading of all detectors of this type differs from the true value 105, so our hypotheses are H0: differs from the true value 105, so our hypotheses are H0: µµ = 105 and = 105 and Ha: Ha: µµ 105. A significance test to answer this question was carried out. 105. A significance test to answer this question was carried out. The test statistic is The test statistic is zz = –0.3336, and the = –0.3336, and the PP-value is 0.74.-value is 0.74.

1)1) Describe what a Type I error would be in this situation.Describe what a Type I error would be in this situation.2)2) Calculate the probability of a Type I error for this problem.Calculate the probability of a Type I error for this problem.3)3) The researchers who carried out the study suspect that the large P-value is 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 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 situation, then determine the probability of a Type II error when in fact µµ = 100. Finally, compute the power of the test against the alternative.= 100. Finally, compute the power of the test against the alternative.

4) If the sample size is increased to4) If the sample size is increased to n n = 30, what will be the power against = 30, what will be the power against the alternative,the alternative, µ µ = 100? = 100? What happened to the power as the sample size What happened to the power as the sample size increased?increased?