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## Chapter 13 Comparing Groups: Analysis of Variance Methods

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**Chapter 13Comparing Groups: Analysis of Variance Methods**• Learn …. How to use Statistical inference To Compare Several Population Means**Section 13.1**How Can We Compare Several Means? One-Way ANOVA**Analysis of Variance**• The analysis of variance method compares means of several groups • Let g denote the number of groups • Each group has a corresponding population of subjects • The means of the response variable for the g populations are denoted by µ1,µ2, … µg**Hypotheses and Assumptions for the ANOVA Test**• The analysis of variance is a significance test of the null hypothesis of equal population means: • H0: µ1 =µ2 = …= µg • The alternative hypothesis is: • Ha:At least two of the population means are unequal**Hypotheses and Assumptions for the ANOVA Test**The assumptions for the ANOVA test comparing population means are as follows: • The population distributions of the response variable for the g groups are normal with the same standard deviation for each group**Hypotheses and Assumptions for the ANOVA Test**• Randomization: • In a survey sample, independent random samples are selected from the g populations • In an experiment, subjects are randomly assigned separately to the g groups**Example: How Long Will You Tolerate Being Put on Hold?**• An airline has a toll-free telephone number for reservations • The airline hopes a caller remains on hold until the call is answered, so as not to lose a potential customer**Example: How Long Will You Tolerate Being Put on Hold?**• The airline recently conducted a randomized experiment to analyze whether callers would remain on hold longer, on the average, if they heard: • An advertisement about the airline and its current promotion • Muzak (“elevator music”) • Classical music**Example: How Long Will You Tolerate Being Put on Hold?**• The company randomly selected one out of every 1000 calls in a week • For each call, they randomly selected one of the three recorded messages • They measured the number of minutes that the caller stayed on hold before hanging up (these calls were purposely not answered)**Example: How Long Will You Tolerate Being Put on Hold?**• Denote the holding time means for the populations that these three random samples represent by: • µ1 = mean for the advertisement • µ2 = mean for the Muzak • µ3 = mean for the classical music**Example: How Long Will You Tolerate Being Put on Hold?**• The hypotheses for the ANOVA test are: • H0: µ1=µ2=µ3 • Ha: at least two of the population means are different**Example: How Long Will You Tolerate Being Put on Hold?**• Here is a display of the sample means:**Example: How Long Will You Tolerate Being Put on Hold?**• As you can see from the graph on the previous page, the sample means are quite different • This alone is not sufficient evidence to enable us to reject H0**Variability between Groups and within Groups**• The ANOVA method is used to compare population means • It is called analysis of variance because it uses evidence about two types of variability**Variability Between Groups and Within Groups**• Two examples of data sets with equal means but unequal variability:**Variability Between Groups and Within Groups**• Which case do you think gives stronger evidence against H0:µ1=µ2=µ3? • What is the difference between the data in these two cases?**Variability Between Groups and Within Groups**• In both cases the variability between pairs of means is the same • In ‘Case b’ the variability within each sample is much smaller than in ‘Case a.’ • The fact that ‘Case b’ has less variability within each sample gives stronger evidence against H0**ANOVA F-Test Statistic**• The analysis of variance (ANOVA) F-test statistic is: • The larger the variability between groups relative to the variability within groups, the larger the F test statistic tends to be**ANOVA F-Test Statistic**• The test statistic for comparing means has the F-distribution • The larger the F-test statistic value, the stronger the evidence against H0**ANOVA F-test for Comparing Population Means of Several**Groups • Assumptions: • Independent random samples • Normal population distributions with equal standard deviations**ANOVA F-test for Comparing Population Means of Several**Groups • Hypotheses: • H0:µ1=µ2= … =µg • Ha:at least two of the population means are different**ANOVA F-test for Comparing Population Means of Several**Groups • Test statistic: F- sampling distribution has df1= g -1, df2 = N – g = (total sample size – no. of groups)**ANOVA F-test for Comparing Population Means of Several**Groups • P-value: Right-tail probability above the observed F- value • Conclusion: If decision is needed, reject if P-value ≤ significance level (such as 0.05)**The Variance Estimates and the ANOVA Table**• Let σ denote the standard deviation for each of the g population distributions • One assumption for the ANOVA F-test is that each population has the same standard deviation, σ**The Variance Estimates and the ANOVA Table**• The F-test statistic is the ratio of two estimates of σ2, the population variance for each group • The estimate of σ2 in the denominator of the F-test statistic uses the variability within each group • The estimate of σ2 in the numerator of the F-test statistic uses the variability between each sample mean and the overall mean for all the data**The Variance Estimates and the ANOVA Table**• Computer software displays the two estimates ofσ2 in the ANOVA table • The MS column contains the two estimates, which are called mean squares • The ratio of the two mean squares is the F- test statistic • This F- statistic has a P-value**Example: ANOVA for Customers’ Telephone Holding Times**• This example is a continuation of a previous example in which an airline conducted a randomized experiment to analyze whether callers would remain on hold longer, on the average, if they heard: • An advertisement about the airline and its current promotion • Muzak (“elevator music”) • Classical music**Example: ANOVA for Customers’ Telephone Holding Times**• Denote the holding time means for the populations that these three random samples represent by: • µ1 = mean for the advertisement • µ2 = mean for the Muzak • µ3 = mean for the classical music**Example: ANOVA for Customers’ Telephone Holding Times**• The hypotheses for the ANOVA test are: • H0:µ1=µ2=µ3 • Ha:at least two of the population means are different**Example: ANOVA for Customers’ Telephone Holding Times**• Since P-value < 0.05, there is sufficient evidence to reject H0:µ1=µ2=µ3 • We conclude that a difference exists among the three types of messages in the population mean amount of time that customers are willing to remain on hold**Assumptions for the ANOVA F-Test and the Effects of**Violating Them • Population distributions are normal • Moderate violations of the normality assumption are not serious • These distributions have the same standard deviation, σ • Moderate violations are also not serious**Assumptions for the ANOVA F-Test and the Effects of**Violating Them • You can construct box plots or dot plots for the sample data distributions to check for extreme violations of normality • Misleading results may occur with the F-test if the distributions are highly skewed and the sample size N is small**Assumptions for the ANOVA F-Test and the Effects of**Violating Them • Misleading results may also occur with the F-test if there are relatively large differences among the standard deviations (the largest sample standard deviation being more than double the smallest one)**The 1998 General Social Survey asked subjects how many**friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61 • State the null hypothesis • The population means for all 12 Zodiac signs are the same. • At least two population means are different.**The 1998 General Social Survey asked subjects how many**friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61 • State the alternative hypothesis • The population means for all 12 Zodiac signs are the same. • At least two population means are different.**The 1998 General Social Survey asked subjects how many**friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61 • Based on what you know about the F distribution would you guess that the test value of 0.61 provides strong evidence against the null hypothesis? • No • Yes**The 1998 General Social Survey asked subjects how many**friend they have. Is this associated with the respondent’s astrological sign (the 13 symbols of the Zodiac)? The ANOVA table for the data reports F=0.61 • The P-value associated with the F-statistic is 0.82. At a significance level of 0.05, what is the correct decision? • Reject Ho • Fail to Reject Ho • Reject Ha • Fail to Reject Ha**Section 13.2**How Should We Follow Up an ANOVA F-Test?**Follow Up to an ANOVA F-Test**• When an analysis of variance F-test has a small P-value, the test does not specify which means are different or how different they are • We can estimate differences between population means with confidence intervals**Confidence Intervals Comparing Pairs of Means**• For two groups i and j, with sample means yi and yj having sample sizes ni and nj, the 95% confidence interval for µi - µj is: • The t-score has df = N – g**Confidence Intervals Comparing Pairs of Means**• Some software refers to this confidence method as the Fisher method • When the confidence interval does not contain 0, we can infer that the population means are different • The interval shows just how different the means may be**Example: Number of Good Friends and Happiness**• A recent GSS study asked: “About how many good friends do you have?” • The study also asked each respondent to indicate whether they were ‘very happy,’ ‘pretty happy,’ or ‘not too happy’**Example: Number of Good Friends and Happiness**• Let the response variable y = number of good friends • Let the categorical explanatory variable x = happiness level**Example: Number of Good Friends and Happiness**• Construct a 95% CI to compare the population mean number of good friends for the two categories: ‘very happy’ and ‘pretty happy.’ • 95% CI formula:**Example: Number of Good Friends and Happiness**• First, use the output to find s: • Use software or a table to find the t-value of 1.963**Example: Number of Good Friends and Happiness**• Next calculate the interval: • Since the CI contains only positive numbers, this suggest that, on average, people who are very happy have more good friends than people who are pretty happy**Controlling Overall Confidence with Many Confidence**Intervals • The confidence interval method just discussed is mainly used when g is small or when only a few comparisons are of main interest • The confidence level of 0.95 applies to any particular confidence interval that we construct