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Elementary Quantitative Skills Kick-Off Exercises – Problems gregor.kastner@fhwn.ac.at Contents 1 “Statistical Common Sense” 1 2 Descriptive Statistics 2 3 Standardized Scores (“Z-Scores”) and the Normal Distribution 6 4 Inferential Statistics 7 Foreword This collection of problems aims at assisting you in critically assessing your own statistical literacy level. To fully profit from the upcoming course, you should be able to solve (almost) all problems from the first three Chapters by the end of the introductory lectures. Moreover, it is strongly advised to be familiar with the idea of inferential statistics and thus, at least conceptually, understand the problems of Chapter 4. If you are unsure regarding certain topics, please refer to material from the Data Analysis Crash Course, your favorite elementary statistics textbook (I personally recommend Triola: Essentials of Statistics) or the main course textbook, Koop: Analysis of Financial/Economic Data. 1 “Statistical Common Sense” Problem 1. Which of the levels of measurement (nominal, ordinal, interval, ratio) is most appropriate? (a) Years of construction of buildings that are for sale in Wiener Neustadt. (b) Austrian area codes (“01” for Vienna, “0732” for Linz, etc.). (c) Types of books according to their cover (“hardcover”, “paperback”, “glossy cover”, “other”). (d) Years in which movies were released. (e) Amount of inhabitants of Austrian provinces. (f) Types of movies (“drama”, “comedy”, “adventure”, “documentary”, etc.). (g) Grades for the QuantEM course – whole numbers from 1 (“Very good”) to 5 (“Unsatisfactory”). A grade ≤ 4 is needed to pass. Problem 2. Identify the sample and the population. Also, determine whether the sample is likely to be representative of the population. (a) The newspaper USA Today published a health survey, and some readers completed the survey and returned it. 1 (b) Some people responded to this request: “Dial 1-900-PRO-LIFE to participate in a telephone poll on abortion. ($1.95 per minute. Average call: 2 minutes. You must be 18 years old.)” (c) A Gallup poll of 1012 randomly surveyed adults found that 9% of them said cloning of humans should be allowed. Problem 3. In the following, use critical thinking to develop an alternative/correct conclusion. (a) Based on a study showing that college graduates tend to live longer than those who do not graduate from college, a researcher concludes that studying causes people to live longer. (b) A study showed that in Orange County, more speeding tickets were issued to minorities than to whites. Conclusion: In Orange County, minorities speed more than whites. (c) Data published in USA Today were used to show that there is a correlation between the number of times songs are played on radio stations and the numbers of times the songs are purchased. Conclusion: Increasing the times that songs are played on radio stations causes sales to increase. Problem 4. (a) Identify a major reason why the mean and median are not meaningful statistics for zip codes (e.g. 2700 for Wr. Neustadt, 1010 for Vienna’s first district, etc.). Find another example where it doesn’t make sense to calculate mean and median. (b) Mean income amongst employees is the income that each employee would receive if the total income were divided equally among everyone who is employed. From data provided by Statistik Austria, the approximate mean income amongst employees can be found for each of Austria’s nine provinces. The means before taxes in e based on the latest data available at the time of this writing are: Burgenland K¨arnten NO¨ OO¨ Salzburg Steiermark Tirol Vorarlberg Wien 28 914 27 423 30 873 28 888 27 118 27 415 25 704 27 925 30 729 Calculate the mean ¯x of all provinces. Does it follow that ¯x is the mean income amongst employees for all of Austria? Why or why not? (c) An editorial, the Poughkeepsie Journal printed this statement: “The median price – the price exactly in between the highest and lowest – . . . ” Does that statement correctly describe the median? Why or why not? (d) Which do you think has more variation: the incomes of a random sample of 1000 adults selected from the general population, or the incomes of a random sample of 1000 statistics teachers? Why?