Hypothesis Testing: Single Mean and Single …

1 Chapter 9 Hypothesis testing : Single Mean andSingle Hypothesis testing : Single Mean and Single Student Learning ObjectivesBy the end of this chapter, the student should be able to: Differentiate between Type I and Type II Errors Describe Hypothesis testing in general and in practice Conduct and interpret Hypothesis tests for a Single population mean, population standard deviationknown. Conduct and interpret Hypothesis tests for a Single population mean, population standard deviationunknown. Conduct and interpret Hypothesis tests for a Single population IntroductionOne job of a statistician is to make statistical inferences about populations based on samples taken from intervalsare one way to estimate a population parameter.

2 Another way to makea statistical inference is to make a decision about a parameter. For instance, a car dealer advertises thatits new small truck gets 35 miles per gallon, on the average. A tutoring service claims that its method oftutoring helps 90% of its students get an A or a B. A company says that women managers in their companyearn an average of $60,000 per statistician will make a decision about these claims. This process is called" Hypothesis testing ."A hy-pothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes adecision as to whether or not the data supports the claim that is made about the this chapter, you will conduct Hypothesis tests on Single means and Single proportions.

3 You will alsolearn about the errors associated with these testing consists of two contradictory hypotheses or statements, a decision based on the data,and a conclusion. To perform a Hypothesis test, a statistician will:1 This content is available online at <http:// >.369370 CHAPTER 9. Hypothesis testing : Single MEAN AND SINGLEPROPORTION1. Set up two contradictory Collect sample data (in homework problems, the data or summary statistics will be given to you).3. Determine the correct distribution to perform the Hypothesis Analyze sample data by performing the calculations that ultimately will support one of the Make a decision and write a meaningful : To do the Hypothesis test homework problems for this chapter and later chapters, makecopies of the appropriate special solution sheets.

4 See the Table of Contents topic "Solution Sheets". Null and Alternate Hypotheses2 The actual test begins by considering twohypotheses. They are called thenull hypothesisand thealternatehypothesis. These hypotheses contain opposing :The null Hypothesis :It is a statement about the population that will be assumed to be true unless itcan be shown to be incorrect beyond a reasonable :The alternate Hypothesis :It is a claim about the population that is contradictory toHoand what weconclude when we : No more than 30% of the registered voters in Santa Clara County voted in the primary : More than 30% of the registered voters in Santa Clara County voted in the primary want to test whether the average grade point average in American colleges is (out of )or : = : 6= want to test if college students take less than five years to graduate from college, on the : 5Ha.

5 <5 Example an issue ofU. S. News and World Report, an article on school standards stated that about halfof all students in France, Germany, and Israel take advanced placement exams and a third same article stated that of U. S. students take advanced placement exams and % if the percentage of U. S. students who take advanced placement exams is more than :p= :p> the null and alternate hypotheses are contradictory, you must examine evidence to decide whichhypothesis the evidence supports. The evidence is in the form of sample data. The sample might supporteither the null Hypothesis or the alternate Hypothesis but not you have determined which Hypothesis the sample supports, you make are twooptions for a decision.

6 They are "rejectHo" if the sample information favors the alternate Hypothesis or2 This content is available online at <http:// >.371"do not rejectHo" if the sample information favors the null Hypothesis , meaning that there is not enoughinformation to reject the Symbols Used inHoandHa:HoHaequal (=)not equal (6=)orgreater than (>)orless than (<)greater than or equal to ( )less than (<)less than or equal to ( )more than (>)Table :Hoalways has a symbol with an equal in has a symbol with an equal in it. Thechoice of symbol depends on the wording of the Hypothesis test. However, be aware that manyresearchers (including one of the co-authors in research work) use=in the Null Hypothesis , evenwith>or<as the symbol in the Alternate Hypothesis .

7 This practice is acceptable because weonly make the decision to reject or not reject the Null Optional Collaborative Classroom ActivityBring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles fromwhich your group can write a null and alternate hypotheses. Discuss your hypotheses with the rest of Outcomes and the Type I and Type II Errors3 When you perform a Hypothesis test, there are four outcomes depending on the actual truth (or falseness)of the null hypothesisHoand the decision to reject or not. The outcomes are summarized in the followingtable:ACTIONHoIS not rejectHoCorrect OutcomeType II errorRejectHoType I ErrorCorrect OutcomeTable four outcomes in the table are: The decision is tonot rejectHowhen, in fact,Hois true (correct decision).

8 The decision is torejectHowhen, in fact,Hois true(incorrect decision known as aType I error). The decision is tonot rejectHowhen, in fact,Hois false(incorrect decision known as aType II error). The decision is torejectHowhen, in fact,Hois false(correct decisionwhose probability is called thePower of the Test).3 This content is available online at <http:// >.372 CHAPTER 9. Hypothesis testing : Single MEAN AND SINGLEPROPORTIONEach of the errors occurs with a particular probability. The Greek letters and represent the probabilities. = probability of a Type I error =P(Type I error)= probability of rejecting the null Hypothesis when thenull Hypothesis is true. = probability of a Type II error =P(Type II error)= probability of not rejecting the null Hypothesis whenthe null Hypothesis is false.

9 And should be as small as possible because they are probabilities of errors. They are rarely Power of the Test is 1 . Ideally, we want a high power that is as close to 1 as following are examples of Type I and Type II the null Hypothesis ,Ho, is: Frank s rock climbing equipment is I error: Frank concludes that his rock climbing equipment may not be safe when, in fact, itreally is II error: Frank concludes that his rock climbing equipment is safe when, infact, it is not safe. = probabilitythat Frank thinks his rock climbing equipment may not be safe when, in fact, itreally is. = probabilitythat Frank thinks his rock climbing equipment is safe when, in fact, it that, in this case, the error with the greater consequence is the Type II error.

10 (If Frank thinkshis rock climbing equipment is safe, he will go ahead and use it.)Example the null Hypothesis ,Ho, is: The victim of an automobile accident is alive when he arrivesat the emergency room of a I error: The emergency crew concludes that the victim is dead when, in fact, the victim II error: The emergency crew concludes that the victim is alive when, in fact, the victimis dead. = probabilitythat the emergency crew thinks the victim is dead when, in fact, he is really alive= P(Type I error). = probabilitythat the emergency crew thinks the victim is alive when, in fact,he is dead = P(Type II error).The error with the greater consequence is the Type I error. (If the emergency crew thinks the victimis dead, they will not treat him.)