Agresti/Franklin Statistics, 1 of 122 Chapter 8 Statistical inference: Significance Tests About Hypotheses Learn …. To use an inferential method called a Significance Test To analyze evidence that data provide To make decisions based on data

Agresti/Franklin Statistics, 2 of 122 Two Major Methods for Making Statistical Inferences about a Population Confidence Interval Significance Test

Agresti/Franklin Statistics, 3 of 122 Questions that Significance Tests Attempt to Answer Does a proposed diet truly result in weight loss, on the average? Is there evidence of discrimination against women in promotion decisions? Does one advertising method result in better sales, on the average, than another advertising method?

Agresti/Franklin Statistics, 4 of 122  Section 8.1 What Are the Steps For Performing a Significance Test?

Agresti/Franklin Statistics, 5 of 122 Hypothesis A hypothesis is a statement about a population, usually of the form that a certain parameter takes a particular numerical value or falls in a certain range of values The main goal in many research studies is to check whether the data support certain hypotheses

Agresti/Franklin Statistics, 6 of 122 Significance Test A significance test is a method of using data to summarize the evidence about a hypothesis A significance test about a hypothesis has five steps

Agresti/Franklin Statistics, 7 of 122 Step 1: Assumptions A (significance) test assumes that the data production used randomization Other assumptions may include: Assumptions about the sample size Assumptions about the shape of the population distribution

Agresti/Franklin Statistics, 8 of 122 Step 2: Hypotheses Each significance test has two hypotheses: The null hypothesis is a statement that the parameter takes a particular value The alternative hypothesis states that the parameter falls in some alternative range of values

Agresti/Franklin Statistics, 9 of 122 Null and Alternative Hypotheses The value in the null hypothesis usually represents no effect The symbol H o denotes null hypothesis The value in the alternative hypothesis usually represents an effect of some type The symbol H a denotes alternative hypothesis

Agresti/Franklin Statistics, 10 of 122 Null and Alternative Hypotheses A null hypothesis has a single parameter value, such as H o : p = 1/3 An alternative hypothesis has a range of values that are alternatives to the one in H o such as H a : p ≠ 1/3 or H a : p > 1/3 or H a : p < 1/3

Agresti/Franklin Statistics, 11 of 122 Step 3: Test Statistic The parameter to which the hypotheses refer has a point estimate: the sample statistic A test statistic describes how far that estimate (the sample statistic) falls from the parameter value given in the null hypothesis

Agresti/Franklin Statistics, 12 of 122 Step 4: P-value To interpret a test statistic value, we use a probability summary of the evidence against the null hypothesis, H o First, we presume that H o is true Next, we consider the sampling distribution from which the test statistic comes We summarize how far out in the tail of this sampling distribution the test statistic falls

Agresti/Franklin Statistics, 13 of 122 Step 4: P-value We summarize how far out in the tail the test statistic falls by the tail probability of that value and values even more extreme This probability is called a P-value The smaller the P-value, the stronger the evidence is against H o

Agresti/Franklin Statistics, 14 of 122 Step 4: P-value

Agresti/Franklin Statistics, 15 of 122 Step 4: P-value The P-value is the probability that the test statistic equals the observed value or a value even more extreme It is calculated by presuming that the null hypothesis H is true

Agresti/Franklin Statistics, 16 of 122 Step 5: Conclusion The conclusion of a significance test reports the P-value and interprets what it says about the question that motivated the test

Agresti/Franklin Statistics, 17 of 122 Summary: The Five Steps of a Significance Test 1. Assumptions 2. Hypotheses 3. Test Statistic 4. P-value 5. Conclusion

Agresti/Franklin Statistics, 18 of 122 Is the Statement a Null Hypothesis or an Alternative Hypothesis? In Canada, the proportion of adults who favor legalize gambling is a. Null Hypothesis b. Alternative Hypothesis

Agresti/Franklin Statistics, 19 of 122 Is the Statement a Null Hypothesis or an Alternative Hypothesis? The proportion of all Canadian college students who are regular smokers is less than 0.24, the value it was ten years ago. a. Null Hypothesis b. Alternative Hypothesis

Agresti/Franklin Statistics, 20 of 122  Section 8.4 Decisions and Types of Errors in Significance Tests

Agresti/Franklin Statistics, 21 of 122 Type I and Type II Errors When H 0 is true, a Type I Error occurs when H 0 is rejected When H 0 is false, a Type II Error occurs when H 0 is not rejected

Agresti/Franklin Statistics, 22 of 122 Significance Test Results

Agresti/Franklin Statistics, 23 of 122 An Analogy: Decision Errors in a Legal Trial

Agresti/Franklin Statistics, 24 of 122 P(Type I Error) = Significance Level α Suppose H 0 is true. The probability of rejecting H 0, thereby committing a Type I error, equals the significance level, α, for the test.

Agresti/Franklin Statistics, 25 of 122 P(Type I Error) We can control the probability of a Type I error by our choice of the significance level The more serious the consequences of a Type I error, the smaller α should be

Agresti/Franklin Statistics, 26 of 122 Type I and Type II Errors As P(Type I Error) goes Down, P(Type II Error) goes Up The two probabilities are inversely related

Agresti/Franklin Statistics, 27 of 122 A significance test about a proportion is conducted using a significance level of The test statistic is The P-value is If H o is true, for what probability of a Type I error was the test designed? a..01 b..05 c d..02

Agresti/Franklin Statistics, 28 of 122 A significance test about a proportion is conducted using a significance level of The test statistic is The P-value is If this test resulted in a decision error, what type of error was it? a. Type I b. Type II