Introduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 9-1

Transcription

1 Introduction to Hypothesis Testing 9-1

2 Learning Outcomes Outcome 1. Formulate null and alternative hypotheses for applications involving a single population mean or proportion. Outcome 2. Know what Type I and Type II errors are. Outcome 3. Correctly formulate a decision rule for testing a hypothesis. Outcome 4. Know how to use the test statistic, critical value, and p-value approaches to test a hypothesis. Outcome 5. Compute the probability of a Type II error. 9-2

3 9.1 Hypothesis Tests for Means Hypothesis Testing is an application of statistical inference Is performed in many industries Is a major part of business statistics Provides managers with a structured analytical method for making decisions about population means, proportions, as well as comparing different populations 9-3

4 Formulating the Hypotheses Null Hypothesis The statement about the population parameter that will be assumed to be true during the conduct of the hypothesis test The null hypothesis will be rejected only if the sample data provide substantial contradictory evidence Alternative Hypothesis The hypothesis that includes all population values not included in the null hypothesis 9-4

5 Formulating Hypothesis - Examples Testing the Status Quo The box of cereal has a mean fill of 16 ounces Testing a Research Hypothesis Goodyear s tire will last longer than its competitor s, or more than 60,000 miles Testing a Claim about Population Average waiting time in a medical clinic is less than 15 minutes 9-5

6 Formulating the Null and Alternative Hypothesis 9-6

7 Types of Statistical Errors Statistical Errors Type I Error No Error Type II Error Rejecting the null hypothesis when it is, in fact, true Failing to reject the null hypothesis when it is, in fact, false 9-7

8 Significance Level and Critical Value 9-8

9 Significance Level and Critical Value 9-9

10 Type I and II Errors Relationship Type I and Type II errors cannot happen at the same time Type I error can only occur if H 0 is true Type II error can only occur if H 0 is false If Type I error probability ( ) decreases, then Type II error probability (β) increases 9-10

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12 Decision Rules and Test Statistics Approaches to Conduct the Hypothesis Test Test Statistic: A function of the sampled observations that provides a basis for testing a statistical hypothesis. 9-12

13 One-Tailed Test: A hypothesis test in which the entire rejection region is located in one tail of the sampling distribution Two-Tailed Test: A hypothesis test in which the entire rejection region is split into two tails of the sampling distribution 9-13

14 Types of Hypothesis Tests Variations in Hypothesis Testing 9-14

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16 Recently, research physicians have developed a knee-replacement surgery process they believe will reduce the average patient recovery time. The hospital board will not recommend the new procedure unless there is substantial evidence to suggest that it is better than the existing procedure. The current mean recovery rate for the standard procedure is 142 days, with a standard deviation of 15 days. 9-16

17 Construct the rejection region Compute test statistic: Reach a decision: Draw a conclusion: Do not reject the null hypothesis There is not sufficient evidence to conclude that the new knee replacement procedure results in a shorter average recovery period

18 x ( /2)L x ( /2)U x ( /2)L x ( /2)U 9-18

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20 Construct the rejection region Compute test statistic: Reach a decision: Draw a conclusion: Do not reject the null hypothesis There is not sufficient evidence to reject the null hypothesis

21 p-value Approach p-value: The probability (assuming the null hypothesis is true) of obtaining a test statistic at least as extreme as the test statistic we calculated from the sample. The p-value is also known as the observed significance level. If p-value <, reject H 0 If p-value, do not reject H

22 p-value Approach Adds a degree of significance to the result of the hypothesis test More than just a simple reject Can now determine how strongly you reject or accept The farther the p-value is from, the stronger the decision 9-22

23 Using p-values to Test a Null Hypothesis - Example Company packages salted and unsalted peanuts in 16-ounce sacks. The company s filling process strives for an average fill amount equal to 16 ounces. Using Excel: Because it is two-tailed hypothesis test: Draw a conclusion: 9-23

24 t-test Statistic: To employ the t-distribution, the following assumption needs to be made: The population is normally distributed 9-24

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27 A tire company conducted a test on a new tire design to determine whether the company could make the claim that the mean tire mileage would exceed 60,000 miles. A simple random sample of 100 tires was tested, and the number of miles each tire lasted was recorded. Compute test statistic: Draw a conclusion: 9-27

28 9.2 Hypothesis Tests for a Proportion 9-28

29 Hypothesis Tests for a Proportion Involves categorical values Two possible outcomes Success (possesses a certain characteristic) Failure (does not possesses that characteristic) Examples: Scored shots to the basket per game Defective items manufactured 9-29

30 Hypothesis Tests for a Proportion z ( / 2)L z ( / 2)U p ( / 2)L p ( / 2)U 9-30

31 Hypothesis Tests for a Proportion 9-31

32 Hypothesis Test for a Population Proportion - Example A League is considering increasing the season ticket prices for basketball games. The marketing manager is concerned that some people will terminate their ticket orders if this change occurs. If more than 10% of the season ticket orders would be terminated, the marketing manager does not want to implement the price increase 9-32

33 Estimation and Hypothesis Testing for Two Populations Parameters 10-1

34 Learning Outcomes Outcome 1. Discuss the logic behind and demonstrate the techniques for using impendent samples to test hypotheses and develop interval estimates for the difference between two popular means. Outcome 2. Develop confidence interval estimates and conduct hypothesis tests for the difference between two population means for paired samples Outcome 3. Carry out hypothesis tests and establish interval estimates, using sample data, for the difference between two population proportions

35 10.1 Estimation for Two Population Means Using Independent Samples Independent Samples Samples selected from two or more populations in such a way that the occurrence of values in one sample has no influence on the probability of the occurrence of values in the other sample(s). Estimation for Two Population Means The population standard deviations are known The population standard deviations are unknown 10-35

36 - Variance of population 1 - Variance of population 2 n 1 and n 2 - Sample sizes from populations 1 and

37 z-values for several of the most used confidence levels: 10-37

38 Step 1: Define the population parameter of interest and select independent samples from the two populations Step 2: Specify the desired confidence level Step 3: Compute the point estimate Step 4: Determine the standard error of the sampling distribution Step 5: Determine the critical value, z, from the standard normal table Step 6: Develop the confidence interval estimate 10-38

39 The objective is to estimate the difference in mean time spent per visit for male and female customers of the fitness center. Previous studies indicate that the standard deviation is 11 minutes for males and 16 minutes for females. Develop a 95% confidence interval estimate for the difference in mean times. Step 1: Select independent samples from the two populations Step 2: Specify the desired confidence level Step 3: Compute the point estimate Step 4: Determine the standard error Step 5: Determine the critical value, z Step 6: Develop the confidence interval estimate 10-39

40 Assumptions: The populations are normally distributed The populations have equal variances The samples are independent Confidence interval estimate should be developed using the t-distribution 10-40

41 - Pooled standard deviation t - Critical t-value from the t-distribution, with degrees of freedom equal to n 1 + n

42 Step 1: Define the population parameter of interest and select independent samples from the two populations Step 2: Specify the confidence level Step 3: Compute the point estimate Step 4: Determine the standard error of sampling distribution Step 5: Determine the critical value, t, from the t-distribution table Step 6: Develop a confidence interval 10-42

43 t - Critical t-value from the t-distribution, with degrees of freedom equal to: 10-43

44 Step 1: Define the population parameter of interest and select independent samples from the two populations Step 2: Specify the desired confidence level Step 3: Compute the point estimate Step 4: Determine the standard error of the sampling distribution Step 5: Calculate the degrees of freedom and determine the critical value, t, from the t-distribution table Step 6: Develop the confidence interval estimate 10-44

45 10.2 Hypothesis Tests for Two Population Means Using Independent Samples Some situations require to test whether two populations have equal means or whether one population mean is larger (or smaller) then another These are hypothesis-testing applications Hypothesis Tests for Two Population Means The population standard deviations are known and the samples are independent The population standard deviations are unknown and the samples are independent 10-45

46 The Hypothesis-Testing Process for Two Population Means Step 1: Specify the population parameter of interest Step 2: Formulate the appropriate null and alternative hypotheses. The null hypothesis should contain the equality Variations in Hypothesis Testing 10-46

47 The Hypothesis-Testing Process for Two Population Means 10-47

48 The Hypothesis-Testing Process for Two Population Means - Example One-tailed (lower) Test Example: The company is interested in determining whether there is a difference in the average thickness of brick facing products made at the two plants. Plant 1 Plant

49 Using p-values 10-49

50 t-test Statistic: 10-50

51 The Hypothesis-Testing Process for Two Population Means - Example Two-tailed Test Example: Suppose an independent test agency wishes to conduct a test to determine whether name-brand ink cartridges generate more color pages on average than competing generic ink cartridges. Name-Brand Generic 10-51

52 The Hypothesis-Testing Process for Two Population Means - Example One-tailed (upper) Test Example: The leaders of the study are interested in determining whether there is a difference in mean annual contributions for individuals covered by TSAs and those with 401(k) retirement programs. TSA 401(k) 10-52

53 t-test Statistic: Degrees of Freedom: 10-53

54 10.3 Interval Estimation and Hypothesis Tests for Paired Samples Paired samples are dependent samples Samples that are selected in such a way that values in one sample are matched with the values in the second sample for the purpose of controlling for extraneous factors Examples Testing a new paint mix vs an old one Testing difference in gas mileage comparing regular and premium gas 10-54

55 Interval Estimation and Hypothesis Tests for Paired Samples x 1 and x 2 Values from samples 1 and 2, respectively d n d i i 1 n 10-55

56 Interval Estimation and Hypothesis Tests for Paired Samples s d n ( d i d ) i 1 n

57 Confidence Interval Estimate 10-57

58 Hypothesis Testing for Paired Samples t-test Statistic for Paired-Sample Test: 10-58

59 Hypothesis Testing for Paired Samples Variations in Hypothesis Testing Reject H 0 if t < -t /2 or t > t /2 Reject H 0 if t < -t Reject H 0 if t > t 10-59

60 Hypothesis Testing for Paired Samples - Example Suppose an independent test agency wishes to conduct a test to determine whether name-brand ink cartridges generate more color pages on average than competing generic ink cartridges. The test is conducted using paired samples. This means that the same people will use both types of cartridges, and the pages printed in each case will be recorded. Reject H 0 if t > t 10-60

61 Hypothesis Testing for Paired Samples - Solution User Name-Brand Generic d i The mean paired difference: The standard deviation for the paired differences: The t-test statistic: Decision and conclusion: Do not reject the null hypothesis 10-61

62 10.4 Estimation and Hypothesis Tests for Two Population Proportions 10-62

63 Estimation and Hypothesis Tests for Two Population Proportions Pooled Estimator for Overall Proportion: z-test Statistic for Difference between Population Proportions: 10-63

64 Hypothesis Testing for Two Population Proportions Variations in Hypothesis Testing 10-64

65 Hypothesis Testing for Two Population Proportions - Example A critical component of a handheld hair dryer is the motor-heater unit. Company has recently created a new motor-heater unit with fewer parts than the current unit. Company has decided to test samples of old and new units to see which motor-heater is more reliable. The null hypothesis states that the new motor-heater is no better than the old, or current, motor-heater. New Unit Old Unit 10-65

66 Analysis of Variance 12-1

67 Learning Outcomes Outcome 1. Understand the basic logic of analysis of variance. Outcome 2. Perform a hypothesis test for a single-factor design using analysis of variance manually and with the aid of Excel software. Outcome 3. Conduct and interpret post-analysis of variance pairwise comparison procedures. Outcome 4. Recognize when randomized block analysis of variance is useful and be able to perform analysis of variance on a randomized block design. Outcome 5. Perform analysis of variance on a two-factor design of experiments with replications using Excel and interpret the output

68 12.1 One-Way Analysis of Variance It is a common situation when someone needs to determine whether three or more populations have equal means ANOVA analysis of variance Completely Randomized Design: An experiment that consists of the independent random selection of observations representing each level of one factor 12-68

69 One-Way Analysis of Variance An analysis of variance design in which independent samples are obtained from two or more levels of a single factor for the purpose of testing whether the levels have equal means. Examples: Accident rates for 1 st, 2 nd, and 3 rd shift Expected mileage for five brands of tires 12-69

70 Introduction to One-Way ANOVA Factor: A quantity under examination in an experiment as a possible cause of variation in the response variable Levels: The categories, measurements, or strata of a factor of interest in the current experiment Balanced Design: An experiment has a balanced design if the factor levels have equal sample sizes

71 One-Way ANOVA Assumptions All populations are normally distributed. The population variances are equal. The observations are independent - that is, the occurrence of any one individual value does not affect the probability that any other observation will occur. The data are interval or ratio level

72 Hypotheses of One-Way ANOVA If the null hypothesis is true, the populations have identical distributions sample means for random samples from each population should be close in value The null hypothesis should be rejected only if the sample means are substantially different (some pairs may be the same) 12-72

73 Hypotheses of One-Way ANOVA 12-73

74 Partitioning the Sum of Squares Total Variation (SST): The aggregate dispersion of the individual data values across the various factor levels Within-Sample Variation (SSW): The dispersion that exists among the data values within a particular factor level Between Sample Variation (SSB): Dispersion among the factor sample means 12-74

75 Partitioned Sum of Squares SST - Total sum of squares SSB - Sum of squares between SSW - Sum of squares within 12-75

76 Total Sum of Squares i k n j i

77 Sum of Squares Between k i 1 Variation Due to Differences Among Groups 12-77

78 Sum of Squares Within i k n j i 1 1 Variation Due to Differences Within Groups 12-78

79 Mean Squares Mean Square Between Samples: Mean Square Within Samples: 12-79

80 One-Way ANOVA Table Source of Variation Between Samples Within Samples Total SS df MS F-Ratio SSB MSB SSW MSW SST 12-80

81 One-Way ANOVA - Example Company runs business in several locations. The VP of sales for the company is interested in knowing whether the dollar value for orders made by individual customers differs, on average, between the four locations. Business Locations Mean Variance n

82 One-Way ANOVA - Example Source of Variation SS df MS F-Ratio Between Samples Within Samples Total Draw a conclusion: 12-82

83 One-Way Analysis of Variance 12-83

84 One-Way Analysis of Variance Step 5: Determine the decision rule Step 6: Compute the total sum of squares, sum of squares between, and sum of squares within, and complete the ANOVA table. Step 7: Reach a decision Step 8: Draw a conclusion 12-84

85 Goodness-of-Fit Tests and Contingency Analysis 13-1

86 Learning Outcomes Outcome 1. Utilize the chi-square goodness-of-fit tests to determine whether data from a process fit a specified distribution. Outcome 2. Set up a contingency analysis table and perform a chi-square test of independence

87 13.1 Introduction to Goodness-of-Fit Tests Many of the statistical procedures introduced in earlier chapters require that the sample data come from populations that are normally distributed A statistical technique known as a goodness-of-fit test can be used to find whether the actual data from the process fit the probability distribution being considered

88 Introduction to Goodness-of-Fit Tests Examples: Are technical support calls equal across all days of the week? (i.e., do calls follow a uniform distribution?) Do measurements from a production process follow a normal distribution? 13-88

89 Chi-Square Goodness-of-Fit Test Statistic k i

90 Chi-Square Goodness-of-Fit Test - Example The company runs the same number of taxis Monday through Friday, with reduced staffing on Saturday and Sunday. This is because the operations manager believes that demand for taxicabs is fairly level throughout the week and about 25% less on weekends. The manager has decided to study demand to see whether the assumed demand pattern still applies

91 Chi-Square Goodness-of-Fit Test - Example Data: Test Statistic: Conclusion: 13-91

92 Chi-Square Goodness-of-Fit Test Step 1: Formulate the appropriate null and alternative hypotheses Step 2: Specify the significance level Step 3: Determine the critical value Step 4: Collect the sample data and compute the chi-square test statistic Step 5: Reach a decision Step 6: Draw a conclusion 13-92

93 Normal Distribution Example Company makes wood moldings, doorframes, and window frames. First they rip lumber in a smaller strips. The manufacturer of the saw claims that the ripsaw cuts an average deviation of zero from target and that the differences from target will be normally distributed, with a standard deviation of 0.01 inch. Company has recently become concerned that the ripsaw may not be cutting to the manufacturer s specifications. A quality improvement team selected a random sample of 300 boards just as they came off the ripsaw

94 Normal Distribution Example Conclusion: In this case, because the null hypothesis specified both the mean and the standard deviation, the normal distribution probabilities were computed using these values. If the mean and/or the standard deviation had not been specified, the sample mean and standard deviation would be used in the probability computation. You would lose one additional degree of freedom for each parameter that was estimated from the sample data

95 Chi-Square Goodness-of-Fit Test Step 1: Formulate the appropriate null and alternative hypotheses Step 2: Specify the level of significance Step 3: Determine the critical value Step 4: Collect the sample data and compute the chi-square test statistic Step 5: Reach a decision Step 6: Draw a conclusion 13-95

96 13.2 Introduction to Contingency Analysis Situations involving multiple population proportions Categorical data Used to classify sample observations according to two or more characteristics Chi-Square is used to determine independence of the characteristics of interest Data summarized in a contingency table 13-96

97 Contingency Table A table used to classify sample observations according to two or more identifiable characteristics It is also called a cross-tabulation table Example: Source of Funding vs. Gender (two variables) Source of Funding Gender Private State Male Female

98 Chi-Square Contingency Test Statistic r c i 1 j

99 Chi-Square Contingency Test - Example Contingency Table: Source of Funding Gender Private State Male 57 Female 164 Test Statistic: Conclusion: 13-99

100 2 x 2 Contingency Analysis Step 1: Specify the null and alternative hypotheses Step 2: Determine the significance level Step 3: Determine the critical value Step 4: Collect the sample data and compute the chi-square test statistic Step 5: Reach a decision Step 6: Draw a conclusion

101 r x c Contingency Table - Example Company pays market wages, provides competitive benefits, and offers attractive options for employees. However, several supervisors have complained that employee absenteeism is becoming a problem. In response to these complaints, the human resources manager studied a random sample of 500 employees. One aim of this study was to determine whether there is a relationship between absenteeism and marital status. Absenteeism during the past year was broken down into three levels: 0 absences; 1 to 5 absences; over 5 absences. Marital status was divided into four categories: Single; Married; Divorced; Widowed

102 r x c Contingency Table - Example Contingency Table: Absentee Rate Marital Status Over 5 Row Totals Single Married Divorced Widowed Column Totals Test Statistic: Conclusion:

103 r x c Contingency Table - Example 1. Open file. 2. Compute expected cell frequencies using Excel formula. 3. Compute chisquare statistic using Excel formula

104 Chi-square Test Limitations The chi-square distribution is only an approximation for the true distribution for contingency analysis The approximation is quite good when all expected cell frequencies are at least 5.0 When expected cell frequencies drop below 5.0, the calculated chi-square value tends to be inflated and may inflate the true probability of a Type I error beyond the stated significance level

105 Chi-square Test Limitations There are two alternatives that can be used to overcome the small expected-cellfrequency problem: Increase the sample size Combine the categories of the row and/or column variables

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