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Chapter 13 Comparing Groups: Analysis of Variance Methods

Chapter 13 Comparing Groups: Analysis of Variance Methods

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Chapter 13 Comparing Groups: Analysis of Variance Methods

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  1. Chapter 13Comparing Groups: Analysis of Variance Methods • Learn …. How to use Statistical inference To Compare Several Population Means

  2. Section 13.1 How Can We Compare Several Means? One-Way ANOVA

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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

  9. 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)

  10. Example: How Long Will You Tolerate Being Put on Hold?

  11. 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

  12. 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

  13. Example: How Long Will You Tolerate Being Put on Hold? • Here is a display of the sample means:

  14. 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

  15. 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

  16. Variability Between Groups and Within Groups • Two examples of data sets with equal means but unequal variability:

  17. 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?

  18. 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

  19. 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

  20. 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

  21. ANOVA F-test for Comparing Population Means of Several Groups • Assumptions: • Independent random samples • Normal population distributions with equal standard deviations

  22. 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

  23. 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)

  24. 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)

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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

  31. Example: ANOVA for Customers’ Telephone Holding Times

  32. 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

  33. 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

  34. 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

  35. 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)

  36. 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.

  37. 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.

  38. 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

  39. 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

  40. Section 13.2 How Should We Follow Up an ANOVA F-Test?

  41. 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

  42. 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

  43. 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

  44. 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’

  45. Example: Number of Good Friends and Happiness • Let the response variable y = number of good friends • Let the categorical explanatory variable x = happiness level

  46. Example: Number of Good Friends and Happiness

  47. 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:

  48. 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

  49. 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

  50. 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