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Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer

Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer. Chapter 8 Hypothesis Testing : An Introduction. Doing Statistics for Business. Chapter 8 Objectives What Is a Hypothesis Test? Overview of Hypotheses to be Tested

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Doing Statistics for Business Data, Inference, and Decision Making Marilyn K. Pelosi Theresa M. Sandifer

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  1. DoingStatistics for BusinessData, Inference, and Decision MakingMarilyn K. PelosiTheresa M. Sandifer Chapter 8 Hypothesis Testing : An Introduction

  2. DoingStatistics for Business Chapter 8 Objectives • What Is a Hypothesis Test? • Overview of Hypotheses to be Tested • The Pieces of a Hypothesis Test • Two-Tail Tests of the Mean: Large Sample • Which Theory Should Go into the Null Hypothesis?

  3. DoingStatistics for Business Chapter 8 Objectives (con’t) • One-Tail Tests of the Mean: Large Sample • What Error Could You be Making?

  4. DoingStatistics for Business What is an Hypothesis Test? The word hypothesis has the same meaning in statistics as it does in everyday use. What does this word mean to you? Some possibilities are: • an idea • an assumption • a guess • a theory

  5. DoingStatistics for Business In statistics, a Hypothesis is an idea, an assumption, or a theory about the characteristics of one or more populations.

  6. DoingStatistics for Business A Hypothesis Test is a statistical procedure that involves formulating a hypothesis and using sample data to decide on the validity of the hypothesis.

  7. DoingStatistics for Business TRY IT NOW! 7-11 StoresPossible Hypotheses 7-11 stores are located all over the Northeast. Management is studying sales data. Develop a specific hypothesis for each of the different types of hypotheses that we have discussed. Here’s one hypothesis to get you started: Type of Hypothesis Specific Hypothesis The shape of the distribution of the variable Daily sales at Store 23 are in one population normally distributed

  8. DoingStatistics for Business TRY IT NOW! The Sports Complex Possible Hypotheses A particular university is considering building a new sports complex. It wishes to know if the sports complex would be widely used by students. Develop a specific hypothesis for each of the different types of hypotheses that we have discussed.

  9. Doing Statistics for Business The Null Hypothesis is a statement about a parameter of the population(s). It is referred to as H0. The Alternative Hypothesis is a statement about a parameter of the population(s) that is opposite to the null hypothesis. It is referred to as HA.

  10. DoingStatistics for Business TRY IT NOW! The Potato Chip Manufacturer Setting Up the Null and Alternative Hypotheses Many people eat chips with their soda. Suppose a potato chip manufacturer is concerned that the bagging equipment may not be functioning properly when filling 10-oz bags. You have been asked to set up a hypothesis test that will help determine if there is a problem with the bagging equipment. What null and alternative hypothesis would you use?

  11. Doing Statistics for Business Discovery Exercise 8.1 Formulating Hypotheses Consider the population of all M & M packages like the one you have in your hand. Step 1: Identify as many different variables as you can. Be sure you have some quantitative and some qualitative variables. Record the values of these variables for your package. (Hint: You should carefully examine the package before you rip it open.) Step2: Select one of the quantitative variables and set up a null and alternative hypothesis for a parameter of this variable.

  12. Doing Statistics for Business Discovery Exercise 8.1 Formulating Hypotheses (con’t) Step 3: Select one of the qualitative variables and set up a null and alternative hypothesis for a parameter of this variable. Step 4: As a class agree on several quantitative and qualitative variables that you feel are important. Record the data for all of the teams on each of these variables. Step 5: Use the tools of descriptive statistics to display these data. Step 6: Enjoy the M&M’s!

  13. Doing Statistics for Business A Test Statistic is calculated from the sample data and is used to decide between the null and alternative hypothesis. The rejection region is the range of values of the test statistic that will lead you to reject the null hypothesis. Alpha,  , is the area of the rejection region.

  14. Doing Statistics for Business 5-Step Hypothesis Testing Procedure Step 1: Set up the null and alternative hypotheses. Step 2: Define the test procedure. This includes selecting the right test, picking the value of , and finding the rejection region. Step 3: Collect the data and calculate the test statistic. Step 4: Decide whether or not to reject the null hypothesis. Step 5: Interpret the statistical decision in terms of the stated problem.

  15. Doing Statistics for Business Figure 8.1 Possible Rejection Region

  16. Doing Statistics for Business Figure 8.2 Rejection Region for a two-tail test of m -z a/2 z a/2

  17. DoingStatistics for Business TRY IT NOW! The Tissue Company Finding the Rejection Region Suppose the tissue company decided to set  at 0.10. Find the rejection region.

  18. Doing Statistics for Business Discovery Exercise 8.2 Exploring the Impact of Varying the Value of  You may remember that in Chapter 4 we examined the data collected by a company that was concerned about the length of time its customers were on hold. The data shown below were first presented to you in Example 4.15 in your textbook. There are 50 observations on customer hold times:

  19. Doing Statistics for Business Discovery Exercise 8.2 Exploring the Impact of Varying the Value of  (con’t) The company wishes to test the hypothesis that the true mean hold time is 7 minutes. The standard deviation of hold times is known to be 2 minutes. (a) Test this hypothesis using a value of  = 0.10. (b) Now vary the value of  and complete the following table.  Rejection Region Decision 0.10 (from part a above) 0.05 0.02 0.01

  20. Doing Statistics for Business Discovery Exercise 8.2 Exploring the Impact of Varying the Value of  (con’t) ( c) What happened to your decision as the value of  changed?

  21. Doing Statistics for Business The p value is defined to be the smallest value of  for which you can reject H0.

  22. DoingStatistics for Business TRY IT NOW! The Tissue Company Finding the p Value Find the p value for the tissue company’s two-tail test of . Recall that the average MDStrength was found to be 980 lb/ream and the Z statistic was calculated to be -2.40.

  23. Doing Statistics for Business A Type I Error is made when you reject the null hypothesis and the null hypothesis is actually true. In other words, you incorrectly reject a true null hypothesis. A Type II Error is made when you fail to reject the null hypothesis and the null hypothesis is false. In other words, you continue to believe a false null hypothesis.

  24. DoingStatistics for Business TRY IT NOW! The Potato Chip Manufacturer Examining the Type I and Type II Errors Find the Type I and Type II errors for the hypothesis test that you set up for the potato chip manufacturer of this chapter.

  25. Doing Statistics for Business The probability of making a Type I Error is called  (alpha). The probability of making a Type II Error is called  (beta).

  26. DoingStatistics for Business TRY IT NOW! New Package Design Setting the Value of  We have seen that a one-sided test is often used to investigate whether a new method of advertising or producing something is better than the existing method. Consider a company that is trying a new package design for its product. The average sales for this product are currently $1500/month. The null and alternative hypotheses would be H0.:   $1500 HA.:  > $1500

  27. DoingStatistics for Business TRY IT NOW! New Package Design Setting the Value of  (con’t) In terms of the company’s decision to adopt or not adopt this new design, what are the Type I and Type II errors?

  28. Doing Statistics for Business A Two-Tail Test of the population mean has the following null and alternative hypotheses: H0:  = [a specific number] HA:   [a specific number]

  29. DoingStatistics for Business TRY IT NOW! The Chapperel Steel Company Setting Up the Null and Alternative Hypotheses for a Two-Tail Test of the Mean Another recent management approach is to have employees become actual partners of the business. Chapperel Steel Company has done exactly this and the company feels that one of the benefits of this concept is that the average number of sick days will decrease. Prior to implementing this program, Chapperel had an average of 7.2 sick days per employee. Set up the null and alternative hypothesis to test if the average number of sick days per employee is different from 7.2.

  30. Doing Statistics for Business A Lower-Tail Test of the population mean has the following null and alternative hypotheses: H0:   [a specific number] HA:  < [a specific number]

  31. DoingStatistics for Business TRY IT NOW! The Bank Example Lower-Tail Test Suppose a bank knows that their customers are waiting in line an average of 10.2 minutes during the lunch hour. The branch manager has decided to add an additional teller during the 12-2 p.m. period and wishes to test the hypothesis that the average wait has decreased due to the additional teller. Set up the null and alternative hypothesis for the bank manager.

  32. Doing Statistics for Business An Upper-Tail Test of the population mean has the following null and alternative hypotheses: H0:   [a specific number] HA:  > [a specific number]

  33. DoingStatistics for Business TRY IT NOW! New Advertising Program Setting Up an Upper-Tail Test Suppose a company has implemented a new advertising program in the hope of increasing sales from last year’s annual average of $4.3 million. Set up the null and alternative hypotheses to test the theory that sales have increased.

  34. DoingStatistics for Business Summary A Two-Tail Test • Is used to test if the parameter has shifted away from a certain number in either direction, increased or decreased. • Must always be set up so the “=“ theory is the null hypothesis. • Is used when the problem statement has the key words changed or different in the problem statement.

  35. DoingStatistics for Business Summary A Lower-Tail Test • Is used to test if the parameter has shifted to a number less than a certain number. • Must always be set up so the “=“ as part of the null hypothesis. • Is used when the problem statement has the key words decreased, reduced, less than. • The theory that you wish to “prove” is placed into the alternative hypothesis.

  36. DoingStatistics for Business Summary An Upper-Tail Test • Is used to test if the parameter has shifted to a number more than a certain number. • Must always be set up so the “=“ as part of the null hypothesis. • Is used when the problem statement has the key words increased, greater than. • The theory that you wish to “prove” is placed into the alternative hypothesis.

  37. DoingStatistics for Business TRY IT NOW! Judicial System Setting Up the Null and Alternative Hypothesis If you think about the judicial system in terms of a hypothesis test, how would you set up the null and the alternative hypotheses?

  38. DoingStatistics for Business TRY IT NOW! VCR Manufacturer Setting Up the Hypotheses So the Status Quo Is in the Null Hypothesis Suppose a manufacturer of VCRs claims that the average life of his VCRs is at least three years. You have a VCR made by this company and have had problems with it, and so you question this claim. Set up the hypothesis test to investigate the manufacturers claims.

  39. DoingStatistics for Business TRY IT NOW! Finding the Rejection Region Try to predict what will happen to the -Z a value for the one-tail test if  = 0.025. Now find it to confirm your guess.

  40. DoingStatistics for Business TRY IT NOW! Frozen Foods Lower-Tail Test of the Mean Jake Bramhall can identify the make, model, and number of cylinders of any passing car but he can’t tell the difference between stewed tomatoes and tomato paste. While more men are pushing shopping carts these days, many like Mr.. Bramhall show little aptitude in the supermarket and display markedly different purchasing behavior than women.

  41. DoingStatistics for Business TRY IT NOW! Frozen Foods Lower-Tail Test of the Mean (con’t) A study done by Consumer Network Inc. shows that the average amount of money spent by 100 single men on facial tissues was $7.38. On the basis of these data can you conclude that men spend less money on facial tissues than the average $8.19 spent by women on facial tissues? Use a population standard deviation of $3.50 and  value of 0.05. Are the results different if you use  = 0.01?

  42. DoingStatistics for Business TRY IT NOW! Supermarket Survey Upper-Tail Test of the Mean From the same supermarket survey it is found that the 100 men spent, on on the average, $19.98 on low-calorie soft drinks. Is there enough evidence to conclude that men spend more than women, who, on the average, spent $18.86? Assume that the population standard deviation is $10 and use  = 0.05.

  43. Doing Statistics for Business Figure 8.3 Rejection Region for a two-tail test of m with  = 0.05

  44. Doing Statistics for Business Chapter 8 Summary In this chapter you have learned: • The key steps in doing any Hypothesis Test begins with formulating two opposing viewpoints called the Null and Alternative Hypotheses. • These hypotheses are theories or ideas about the value of one or more population parameters.

  45. Doing Statistics for Business Chapter 8 Summary (con’t) • The technique of Hypothesis Testing helps you decide between these opposing hypotheses using the sample data as the evidence upon which to base your decision. • In doing any hypothesis test there are two possible errors you can make Type I and Type II

  46. Doing Statistics for Business Chapter 8 Summary (con’t) • The probability of making these errors are labeled  and , respectively. • Large Sample Tests are applied whenever you know the population standard deviation or if you have a sufficiently large sample size, n > 30.

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