Introduction to Hypothesis Testing. Copyright 2014 Pearson Education, Inc. 91


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1 Introduction to Hypothesis Testing 91
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 pvalue approaches to test a hypothesis. Outcome 5. Compute the probability of a Type II error. 92
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 93
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 94
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 95
6 Formulating the Null and Alternative Hypothesis 96
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 97
8 Significance Level and Critical Value 98
9 Significance Level and Critical Value 99
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 910
11 911
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. 912
13 OneTailed Test: A hypothesis test in which the entire rejection region is located in one tail of the sampling distribution TwoTailed Test: A hypothesis test in which the entire rejection region is split into two tails of the sampling distribution 913
14 Types of Hypothesis Tests Variations in Hypothesis Testing 914
15 915
16 Recently, research physicians have developed a kneereplacement 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. 916
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 918
19 919
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 pvalue Approach pvalue: 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 pvalue is also known as the observed significance level. If pvalue <, reject H 0 If pvalue, do not reject H
22 pvalue 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 pvalue is from, the stronger the decision 922
23 Using pvalues to Test a Null Hypothesis  Example Company packages salted and unsalted peanuts in 16ounce sacks. The company s filling process strives for an average fill amount equal to 16 ounces. Using Excel: Because it is twotailed hypothesis test: Draw a conclusion: 923
24 ttest Statistic: To employ the tdistribution, the following assumption needs to be made: The population is normally distributed 924
25 925
26 926
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: 927
28 9.2 Hypothesis Tests for a Proportion 928
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 929
30 Hypothesis Tests for a Proportion z ( / 2)L z ( / 2)U p ( / 2)L p ( / 2)U 930
31 Hypothesis Tests for a Proportion 931
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 932
33 Estimation and Hypothesis Testing for Two Populations Parameters 101
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 1035
36  Variance of population 1  Variance of population 2 n 1 and n 2  Sample sizes from populations 1 and
37 zvalues for several of the most used confidence levels: 1037
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 1038
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 1039
40 Assumptions: The populations are normally distributed The populations have equal variances The samples are independent Confidence interval estimate should be developed using the tdistribution 1040
41  Pooled standard deviation t  Critical tvalue from the tdistribution, 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 tdistribution table Step 6: Develop a confidence interval 1042
43 t  Critical tvalue from the tdistribution, with degrees of freedom equal to: 1043
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 tdistribution table Step 6: Develop the confidence interval estimate 1044
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 hypothesistesting 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 1045
46 The HypothesisTesting 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 1046
47 The HypothesisTesting Process for Two Population Means 1047
48 The HypothesisTesting Process for Two Population Means  Example Onetailed (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 pvalues 1049
50 ttest Statistic: 1050
51 The HypothesisTesting Process for Two Population Means  Example Twotailed Test Example: Suppose an independent test agency wishes to conduct a test to determine whether namebrand ink cartridges generate more color pages on average than competing generic ink cartridges. NameBrand Generic 1051
52 The HypothesisTesting Process for Two Population Means  Example Onetailed (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) 1052
53 ttest Statistic: Degrees of Freedom: 1053
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 1054
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 1055
56 Interval Estimation and Hypothesis Tests for Paired Samples s d n ( d i d ) i 1 n
57 Confidence Interval Estimate 1057
58 Hypothesis Testing for Paired Samples ttest Statistic for PairedSample Test: 1058
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 1059
60 Hypothesis Testing for Paired Samples  Example Suppose an independent test agency wishes to conduct a test to determine whether namebrand 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 1060
61 Hypothesis Testing for Paired Samples  Solution User NameBrand Generic d i The mean paired difference: The standard deviation for the paired differences: The ttest statistic: Decision and conclusion: Do not reject the null hypothesis 1061
62 10.4 Estimation and Hypothesis Tests for Two Population Proportions 1062
63 Estimation and Hypothesis Tests for Two Population Proportions Pooled Estimator for Overall Proportion: ztest Statistic for Difference between Population Proportions: 1063
64 Hypothesis Testing for Two Population Proportions Variations in Hypothesis Testing 1064
65 Hypothesis Testing for Two Population Proportions  Example A critical component of a handheld hair dryer is the motorheater unit. Company has recently created a new motorheater unit with fewer parts than the current unit. Company has decided to test samples of old and new units to see which motorheater is more reliable. The null hypothesis states that the new motorheater is no better than the old, or current, motorheater. New Unit Old Unit 1065
66 Analysis of Variance 121
67 Learning Outcomes Outcome 1. Understand the basic logic of analysis of variance. Outcome 2. Perform a hypothesis test for a singlefactor design using analysis of variance manually and with the aid of Excel software. Outcome 3. Conduct and interpret postanalysis 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 twofactor design of experiments with replications using Excel and interpret the output
68 12.1 OneWay 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 1268
69 OneWay 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 1269
70 Introduction to OneWay 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 OneWay 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 OneWay 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) 1272
73 Hypotheses of OneWay ANOVA 1273
74 Partitioning the Sum of Squares Total Variation (SST): The aggregate dispersion of the individual data values across the various factor levels WithinSample 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 1274
75 Partitioned Sum of Squares SST  Total sum of squares SSB  Sum of squares between SSW  Sum of squares within 1275
76 Total Sum of Squares i k n j i
77 Sum of Squares Between k i 1 Variation Due to Differences Among Groups 1277
78 Sum of Squares Within i k n j i 1 1 Variation Due to Differences Within Groups 1278
79 Mean Squares Mean Square Between Samples: Mean Square Within Samples: 1279
80 OneWay ANOVA Table Source of Variation Between Samples Within Samples Total SS df MS FRatio SSB MSB SSW MSW SST 1280
81 OneWay 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 OneWay ANOVA  Example Source of Variation SS df MS FRatio Between Samples Within Samples Total Draw a conclusion: 1282
83 OneWay Analysis of Variance 1283
84 OneWay 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 1284
85 GoodnessofFit Tests and Contingency Analysis 131
86 Learning Outcomes Outcome 1. Utilize the chisquare goodnessoffit tests to determine whether data from a process fit a specified distribution. Outcome 2. Set up a contingency analysis table and perform a chisquare test of independence
87 13.1 Introduction to GoodnessofFit 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 goodnessoffit test can be used to find whether the actual data from the process fit the probability distribution being considered
88 Introduction to GoodnessofFit 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? 1388
89 ChiSquare GoodnessofFit Test Statistic k i
90 ChiSquare GoodnessofFit 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 ChiSquare GoodnessofFit Test  Example Data: Test Statistic: Conclusion: 1391
92 ChiSquare GoodnessofFit 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 chisquare test statistic Step 5: Reach a decision Step 6: Draw a conclusion 1392
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 ChiSquare GoodnessofFit 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 chisquare test statistic Step 5: Reach a decision Step 6: Draw a conclusion 1395
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 ChiSquare is used to determine independence of the characteristics of interest Data summarized in a contingency table 1396
97 Contingency Table A table used to classify sample observations according to two or more identifiable characteristics It is also called a crosstabulation table Example: Source of Funding vs. Gender (two variables) Source of Funding Gender Private State Male Female
98 ChiSquare Contingency Test Statistic r c i 1 j
99 ChiSquare Contingency Test  Example Contingency Table: Source of Funding Gender Private State Male 57 Female 164 Test Statistic: Conclusion: 1399
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 chisquare 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 Chisquare Test Limitations The chisquare 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 chisquare value tends to be inflated and may inflate the true probability of a Type I error beyond the stated significance level
105 Chisquare Test Limitations There are two alternatives that can be used to overcome the small expectedcellfrequency problem: Increase the sample size Combine the categories of the row and/or column variables
106 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America
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