І. Write out the statistical terms from the text and find their Ukrainian equivalents.

When comparing different populations or treatments, the data are subject to sampling variability. For example, we would not expect the same sales data if we mailed various advertising offers to a different sample of households. We therefore pose the question for inference in terms of the mean response.

Weare now ready to extend those methods to problems involving more than two populations.Thestatisticalmethodologyforcomparingseveralmeansiscalled analysis of variance,orsimplyANOVA.Inthe sectionsthatfollow, wewill examine the basic ideas and assumptions that are needed for ANOVA. Although the details differ, many of the concepts are similar to those discussed in the two-sample case.

 WewillconsidertwoANOVAtechniques.Whenthereisonlyonewaytoclassify the populations of interest, we use one-way ANOVA to analyze the data. Forexample,tocomparethesurvivaltimesforthreedifferentlungcancertherapies we use one-way ANOVA.

 Inmanyothercomparisonstudies,thereismorethanonewaytoclassifythe populations. For the advertising study, the company may also consider mailing the offers using two different envelope styles. Will each offer draw more sales on the average when sent in an attention-grabbing envelope? Analyzing the effect of advertising offer and envelope layout together requires two-way ANOVA.

Data for one-way ANOVA

 One-wayanalysisofvarianceisastatisticalmethodforcomparingseveralpopulation means. We draw a simple random sample (SRS) from each population and use the data to test the null hypothesis that the population means are all equal. Consider the following two examples:

EXAMPLE

1.  Choosing the best magazine layout. A magazine publisher wants to compare three different layouts for a magazine that will be offered for sale at supermarket checkout lines. She is interested in whether there is a layout that better catches shoppers’ attention and results in more sales. To investigate, she randomly assigns each of 60 stores to one of the three layouts and records the number of magazines that are sold in a one-week period.

2. Average age of bookstore customers. How do five bookstores in thesamecitydifferinthedemographicsoftheircustomers?Arecertainbookstores morepopularamong teenagers? Doupper-incomeshoppers tendtogo to one store? A market researcher asks 50 customers of each store to respond to a questionnaire. Two variables of interest are the customer’s age and income level.

These two examples are similar in that

• There is a single quantitative response variable measured on many units; the units are stores in the first example and customers in the second.

 • Thegoalistocompareseveral populations:storesdisplayingthreemagazine layouts in the first example and customers of five bookstores in the second.

There is, however, an important difference.

Example 1 describes an experiment, in which stores are randomly assigned to layouts. Example 2 is an observational study in which customers are selected during a particular time periodandnotallagreetoprovidedata.Wewilltreatoursamplesofcustomers as random samples even though this is only approximately true.

In both examples, we will use ANOVA to compare the mean responses. The same ANOVA methods apply to data from random samples and to data from randomized experiments. It is important to keep the data-production method in mind when interpreting the results. A strong case for causation is best made by a randomized experiment.

Comparing means

The question we ask in ANOVA is “Do all groups have the same population mean?” We will often use the term groups for the populations to be compared in a one-way ANOVA. To answer this question we compare the sample means.

 Figure 12.1 displays the sample means for Example 12.1. It appears that Layout 2 has the highest average sales. But is the observed difference in sample means just the result of chance variation? We should not expect sample means to be equal, even if the population means are all identical.

Thepurpose ofANOVAis toassess whether theobserved differencesamong sample means are statistically significant. Could a variation among the three sample means this large be plausibly due to chance, or is it good evidence for a differenceamongthepopulationmeans?Thisquestioncan’tbeansweredfrom the sample means alone. Because the standard deviation of a sample mean x is the population standard deviation σ divided by √n, the answer also dependsupon both the variation within the groups of observations and the sizes of the samples.

An overview of ANOVA

ANOVA tests the null hypothesis that the population means are all equal. The alternative is that they are not all equal. This alternative could be true because all of the means are different or simply because one of them differs from the rest. This is a more complex situation than comparing just two populations. If we reject the null hypothesis, we need to perform some further analysis to draw conclusionsabout which populationmeansdifferfrom whichothers and by how much. ThecomputationsneededforanANOVAaremorelengthythanthoseforthe t test. For this reason we generally use computer programs to perform the calculations. Automating the calculations frees us from the burden of arithmetic and allows us to concentrate on interpretation. Complicated computations do notguaranteeavalidstatisticalanalysis.WeshouldalwaysstartourANOVAwith a careful examination of the data using graphical and numerical summaries.

Vocabulary tasks

І. Write out the statistical terms from the text and find their Ukrainian equivalents.

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