Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 1 FINAL EXAMINATION STUDY MATERIAL III A ADDITIONAL READING MATERIAL – INTRO STATS 3 RD EDITION CHAPTER 20 PARTS OF CHAPTER 22 PARTS OF CHAPTER 23

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. CHAPTER 20 Significance Tests About Hypotheses TESTING HYPOTHESES ABOUT PROPORTIONS

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 3 HYPOTHESIS TESTING ABOUT PROPORTIONS OBJECTIVES DEFINE THE NULL AND ALTERNATIVE HYPOTHESES STATE CONCLUSIONS TO HYPOTHESIS TESTS

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 4 DEFINITION A HYPOTHESIS IS A STATEMENT REGARDING A CHARACTERISTIC OF ONE OR MORE POPULATIONS. HYPOTHESIS TESTING IS A PROCEDURE, BASED ON SAMPLE EVIDENCE AND PROBABILITY, USED TO TEST STATEMENTS REGARDING A CHARACTERISTIC OF ONE OR MORE POPULATION.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 5 PROBLEM FORMULATION SUPPOSE WE TOSSED A COIN 100 TIMES AND WE OBTAINED 38 HEADS AND 62 TAILS. IS THE COIN BIASED? THERE IS NO WAY TO SAY YES OR NO WITH 100% CERTAINTY. BUT WE MAY EVALUATE THE STRENGTH OF SUPPORT TO THE HYPOTHESIS [OR CLAIM] THAT “THE COIN IS BIASED.”

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 6 THE NULL AND ALTERNATIVE HYPOTHESES Null hypothesis: Represented by H 0, is a statement that there is nothing happening. Generally thought of as the status quo, or no relationship, or no difference. Usually the researcher hopes to disprove or reject the null hypothesis. Alternative hypothesis: Represented by H a, is a statement that something is happening. In most situations, it is what the researcher hopes to prove. It may be a statement that the assumed status quo is false, or that there is a relationship, or that there is a difference.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 7 NULL AND ALTERNATIVE HYPOTHESES NULL HYPOTHESIS 1.ESTABLISHED FACT; 2.A STATEMENT THAT WE EXPECT DATA TO CONTRADICT; 3.NO CHANGE OF PARAMETERS. ALTERNATIVE HYPOTHESIS 1.NEW CONJECTURE; 2.YOUR CLAIM; 3.A STATEMENT THAT NEEDS A STRONG SUPPORT FROM DATA TO CLAIM IT; 4.CHANGE OF PARAMETERS

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 8 WRITING NULL AND ALTERNATIVE HYPOTHESES Possible null and alternative hypotheses: 1. H 0 : p = p 0 versus H a : p  p 0 (two-sided) 2. H 0 : p = p 0 versus H a : p < p 0 (one-sided) 3. H 0 : p = p 0 versus H a : p > p 0 (one-sided) p 0 = specific value called the null value.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 9 DEMONSTRATIVE EXAMPLE

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 10 EXAMPLES WRITE THE NULL AND ALTERNATIVE HYPOTHESES YOU WOULD USE TO TEST EACH OF THE FOLLOWING SITUATIONS. (A) IN THE 1950s ONLY ABOUT 40% OF HIGH SCHOOL GRADUATES WENT ON TO COLLEGE. HAS THE PERCENTAGE CHANGED? (B) 20% OF CARS OF A CERTAIN MODEL HAVE NEEDED COSTLY TRANSMISSION WORK AFTER BEING DRIVEN BETWEEN 50,000 AND 100,000 MILES. THE MANUFACTURER HOPES THAT REDESIGN OF A TRANSMISSION COMPONENT HAS SOLVED THIS PROBLEM. (C) WE FIELD TEST A NEW FLAVOR SOFT DRINK, PLANNING TO MARKET IT ONLY IF WE ARE SURE THAT OVER 60% OF THE PEOPLE LIKE THE FLAVOR.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 11 SIGNIFICANCE TESTS BACK TO HYPOTHESIS TESTING ABOUT PROPORTIONS

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 12 Steps of a Significance Test about a Population Proportion Step 1: Assumptions  The variable is categorical  The data are obtained using randomization  The 10% Condition  The sample size is sufficiently large that the sampling distribution of the sample proportion is approximately normal:

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 13 Step 2: Hypotheses The null hypothesis has the form: The alternative hypothesis has the form: Steps of a Significance Test about a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 14 Step 3: Test Statistic The test statistic measures how far the sample proportion falls from the null hypothesis value,, relative to what we’d expect if were true The test statistic is: Steps of a Significance Test about a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 15 Step 3:Test Statistic The z-statistic for the significance test is represents the sample estimate of the proportion; p 0 represents the specific value in null hypothesis; n is the sample size.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 16 Step 4: P-value  The P-value summarizes the evidence.  It describes how unusual the observed data would be if were true. Steps of a Significance Test about a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 17 Computing P - Value For H a less than, find probability the test statistic z could have been equal to or less than what it is. For H a greater than, find probability the test statistic z could have been equal to or greater than what it is. For H a two-sided, p-value includes the probability areas in both extremes of the distribution of the test statistic z.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 18 Computing P - Value

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 19 Step 5: Conclusion We summarize the test by reporting and interpreting the P-value. Steps of a Significance Test about a Population Proportion

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 20 Using the P – Value to Reach a Conclusion The level of significance, denoted by  (alpha), is a value chosen by the researcher to be the borderline between when a p-value is small enough to choose the alternative hypothesis over the null hypothesis, and when it is not. When the p-value is less than or equal to , we reject the null hypothesis. When the p-value is larger than , we cannot reject the null hypothesis. The level of significance may also be called the  -level of the test. Decision: reject H 0 if the p-value is smaller than  (usually 0.05, sometimes 0.10 or 0.01). In this case the result is statistically significant

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 21 Using the P – Value to Reach a Conclusion When the p-value is small, we reject the null hypothesis or, equivalently, we accept the alternative hypothesis. “Small” is defined as a p- value  , where  level of significance (usually 0.05). When the p-value is not small, we conclude that we cannot reject the null hypothesis or, equivalently, there is not enough evidence to reject the null hypothesis. “Not small” is defined as a p-value > ,where  = level of significance (usually 0.05).

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 22 Examples From Practice Sheet

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 23 Possible Decisions in a Test of Significance Table 12.1 Possible Decisions in a Test of Significance.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 24 TWO SAMPLE POPULATIONS HYPOTHESES TESTING ABOUT DIFFERENCE IN TWO PROPORTIONS

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 25 STEPS Step 1 Determine null and alternative hypotheses H 0 : p 1 – p 2 =  versus H a : p 1 – p 2   or H a : p 1 – p 2  Watch how Population 1 and 2 are defined.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 26 STEP 2: Verify Data Condition Samples are independent. Sample sizes are large enough so that – are at least 5 and preferably at least 10.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 27 Step 2: The Test Statistic Under the null hypothesis, there is a common population proportion p. This common value is estimated using all the data as The Standardized Test Statistics

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 28 Step 3: Assume the Null Hypothesis is True, Find the P – value For H a less than, the p-value is the area below z, even if z is positive. For H a greater than, the p-value is the area above z, even if z is negative. For H a two-sided, p-value is 2  area above |z|.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 29 Steps 4 and 5 Decide Whether or Not the Result is Statistically Significant based on p-value and Make a Conclusion in Context Choose a level of significance , and reject H 0 if the p-value is less than (or equal to) .

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 30 Examples From Practice Sheet

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. Chapter 23 Statistical Inference: Significance Tests About Hypotheses Significance Tests About Means

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 32 Steps of a Significance Test About a Population Mean Step 1: Assumptions The variable is quantitative. The data are obtained using randomization. The population distribution is approximately normal. This is most crucial when n is small and is one-sided.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 33 Steps of a Significance Test About a Population Mean Step 2: Hypotheses: The null hypothesis has the form: The alternative hypothesis has the form:

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 34 Step 3: Test Statistic The test statistic measures how far the sample mean falls from the null hypothesis value, as measured by the number of standard errors between them. The test statistic is: Steps of a Significance Test About a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 35 Step 4: P-value The P-value summarizes the evidence. It describes how unusual the data would be if were true. Steps of a Significance Test About a Population Mean

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 36 Summary of P-values for Different Alternative Hypotheses Alternative HypothesisP-value Right-tail probability from t distribution Left-tail probability from t distribution Two-tail probability from t distribution The P-value is a single tail or a two-tail probability depending on whether the alternative hypothesis is one- sided or two-sided.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 37 Steps of a Significance Test About a Population Mean Step 5: Conclusion We summarize the test by reporting and interpreting the P-value.

Copyright © 2013, 2009, and 2007, Pearson Education, Inc. 38 Examples From Practice Sheet