Mount Olive High School Summer Assignment for AP Statistics

Transcription

1 Name: Mount Olive High School Summer Assignment for AP Statistics Students who are responsible for completing this packet: Anyone entering: AP Statistics This investigation will count as a major homework assignment for the first marking period. Statistical Reasoning: Investigating a Claim of Discrimination The purpose of this assignment is to familiarize you with the ideas of statistical thinking before you involve yourself with the details of statistical methods. It is easy to get caught in the trap of doing rather than understanding, of asking how rather than why. You can t do statistics unless you learn the methods, but you must not get so caught up in the details of the methods that you lose sight of what they mean. In this assignment you will learn the basic ideas of exploring data uncovering and summarizing patterns, and making inferences from data deciding whether an observed feature of the data could reasonably be attributed to chance alone. The questions will be graded on work shown, explanations / justifications and correctness. All problems are to be completed or well attempted. Show work on every problem for partial credit; crossed out or erased work will not be graded. Write neatly - If I cannot read it then it will be marked wrong.

2 Westvaco Case Study Section 1: Discrimination in the Workplace: Data Exploration Robert Martin was one of 50 people working in the engineering department of Westvaco s envelope division. One spring, Westvaco s management went through five rounds of planning for a reduction in their work force. In Round 1, they eliminated 11 positions, and they eliminated 9 more in Round 2. By the time the layoffs ended, after all five rounds, only 22 of the 50 workers had kept their jobs. The average age in the department had fallen from 48 to 46. After Martin, age 54, was laid off, he sued Westvaco for age discrimination. Display 1.1 shows the data provided by Westvaco to Martin s lawyers. The statistical analysis in the lawsuit used all 50 employees in the engineering department of the envelope division, with separate analyses for salaried and hourly workers. Each row in Display 1.1 corresponds to one worker, and each column corresponds to a characteristic of the worker: job title, whether hourly or salaried, month and year of birth, month and year of hire, and age at birthday in The next-to-last column (Round) tells how the worker fared in the downsizing: 1 means chosen for layoff in Round 1 of planning for the reduction in force, 2 means chosen in Round 2, and so on for Rounds 3, 4, and 5; 0 means not chosen for layoff. The subjects (or objects) of statistical examination are often called cases. In the rows in Display 1.1, the cases are individual Westvaco employees. Their characteristics, in the columns, are called variables. If you pick a row and read across, you find information about a single case. (For example, Robert Martin, in Row 44, was salaried, was born in September 1937, was hired in October 1967, was chosen for layoff in Round 2, and turned 54 in 1991.) Although reading across might seem the natural way to read the table, in statistics you will often find it useful to pick a column and read down. This gives you information about a single variable as you range through all the cases. For example, pick, read down the column, and notice the variability in the ages. It is variability like this the fact that individuals differ that can make it a challenge to see patterns in data and figure out what they mean. Imagine: If there had been no variability if all the workers had been of just two ages, say 30 and 50, and Westvaco had laid off all the 50-year-olds and kept all the 30-year-olds the conclusion would be obvious and there would be no need for statistics. But real life is more subtle than that. The ages of the laid-off workers varied, as did the ages of the workers retained. Statistical methods were designed to cope with such variability. In fact, you might define statistics as the science of learning from data in the presence of variability. Although the bare fact that the ages vary is easy to see in the data table, the pattern of those ages is not so easy to see. This pattern what the values are and how often each occurs is their distribution. In order to see that pattern, a graph is better than a table. The dotplot in Display 1.2 shows the distribution of the ages of the 36 salaried employees who worked in the engineering department just before the layoffs began.

3 Birth Hire at Birthday Row Job Title Pay Mo Yr Mo Yr Round in Engineering Clerk H Engineering Tech II H Engineering Tech II H Secretary to Engin Manag H Engineering Tech II H Engineering Tech II H Engineering Tech II H Parts Crib Attendant H Engineering Tech II H Engineering Tech II H Technical Secretary H Engineering Tech II H Engineering Tech II H Engineering Tech II H Customer Serv Engineer S Customer Serv Engr Assoc S Design Engineer S Design Engineer S Design Engineer S Design Engineer S Engineering Assistant S Engineering Associate S Engineering Manager S Machine Designer S Packaging Engineer S Prod Spec Printing S Proj Eng Elec S Project Engineer S Project Engineer S Project Engineer S Supv Engineering Serv S Supv Machine Shop S Chemist S Design Engineer S Engineering Associate S Machine Designer S Machine Parts Cont Supv S Prod Specialist S Project Engineer S Chemist S Design Engineer S Electrical Engineer S Machine Designer S Machine Parts Cont Coor S VH Prod Specialist S Printing Coordinator S Prod Dev Engineer S Prod Specialist S VH Prod Specialist S Engineering Associate S Display 1.1 The data in Martin v. Westvaco

4 Westvaco pay = "S" Display 1.2: s of the salaried workers. (Each dot represents a worker; the age is shown by the position of the dot along the scale below it.) Display 1.2 provides some useful information about the variability in the ages, but by itself doesn t tell anything about possible age discrimination in the layoffs. For that, you need to distinguish between those salaried workers who lost their jobs and those who didn t. The dotplot in Display 1.3, which shows those laid off (Job Status = 1) and those retained (Job Status = 0), provides weak evidence for Martin s case. Those laid off generally were older than those who kept their jobs, but the pattern isn t striking. Westvaco pay = "S" Display 1.3: Salaried Workers: ages of those laid off and those retained Display 1.3 shows that most salaried workers who were laid off were age 50 or older. However, this alone doesn t support Martin s case because most of the workers were age 50 or older to begin with. One way to proceed is to make a summary table. The table shown here classifies the salaried workers according to age and whether they were laid off or retained. (Using 50 as the dividing age between younger and older is a somewhat arbitrary, but reasonable, decision.) Laid Off Retained Total Under or Older Total To decide whether Martin has a case, compare the proportion of salaried workers under 50 who were laid off (6 out of 16, or 0.375) with the proportion of those 50 or older who were laid off (12 out of 20, or 0.600). These proportions are quite different, an argument in favor of Martin.

5 Looking at the layoffs of the salaried workers round by round provides further evidence in favor of Martin. The dotplots in Display 1.4 show the ages of the salaried workers laid off and retained by round. Westvaco Round1 laid off retained pay = "S" Westvaco Round2 laid off retained pay = "S" Westvaco Round3 laid off retained pay = "S" Westvaco Round4 laid off retained pay = "S" Westvaco Round5 laid off retained pay = "S" Display 1.4: Salaried workers: ages of those laid off (gray circles) and those retained (blue squares) in each round

6 You might feel as if the analysis so far ignores important facts, such as worker qualifications. That s true. However, the first step is to decide whether, based on the data in Display 1.1, older workers were more likely to be laid off. If not, Martin s case fails. If so, it is then up to Westvaco to justify its actions. Exercises E1. This summary table classifies salaried workers as to whether they were laid off and their age, this time using 40 as the cutoff between younger and older workers. Laid Off Retained Total Under or Older Total a. What proportion of workers age 40 or older was laid off? What proportion of laid-off workers was age 40 or older? b. What proportion of workers under age 40 was laid off? What proportion was not laid off? c. What two proportions should you compute and compare in order to decide whether older workers were disproportionately laid off? Make these computations and give your decision. d. Compare this table with the table for the salaried workers shown above, where 50 was the age cutoff. If you were Martin s lawyer, would you present a table using 40 or 50 as the cutoff? E2. Explore whether hourly workers at Westvaco were more likely than salaried workers to lose their jobs. a. Start by constructing a summary table to display the relevant data. b. Compute two proportions that will allow you to make this comparison. c. What do you conclude form comparing the proportions? E3. Twenty-two workers kept their jobs. Explore whether the age distributions are similar for the hourly and salaried workers who kept their jobs. a. Show the two age distributions on a pair of dotplots that have the same scale. How do these distributions differ? b. Do your dotplots in part a support a claim that Westvaco was more inclined to keep older workers if they were salaried rather than hourly?

7 E4. Consider these three facts from your work so far: Salaried workers were more likely to keep their jobs than were hourly workers. Older workers were more likely to be laid off than were younger workers. Older workers were more likely to be salaried than were younger workers. Putting these three facts together, what can you conclude? Is this evidence in favor of Martin s case, or does it help Westvaco? E5. Refer to Display 1.1. a. Create a summary table whose five cases are Round 1 through Round 5 and whose three variables are total number of employees laid off in that round, number of employees laid off in that round who were 40 or older, and percentage laid off in that round who were 40 or older b. Describe any patterns you find in the table and what you think they might mean. E6. Last hired, first fired is shorthand for When you have to downsize, start by laying off the newest person, then the person hired next before that, and work back in reverse order of seniority. (The person who s been working longest will be the last to be laid off.) Examine the Westvaco data. a. How was seniority related to the decisions about layoffs in the engineering department at Westvaco? b. What explanation(s) can you suggest for any patterns you find?

8 E7. Many tables in the media are arranged with cases as rows and variables as columns. For Displays 1.7 and 1.8 in parts a and b, identify the cases and the variables. Then compute the values missing from each table. a. Player Season Team Games At Bats Hits HRs SBs BA H. Nicol 1887 CN R. Henderson 1982 OAK A. Latham 1887 SL L. Brock 1974 STL C. Comiskey 1887 SL b. Stock Closing Price Closing Price Change Percentage Volume on 10/28 on 10/30 Change Chrysler ½ -6 1/ % 269,100 Coca-Cola /8 14,100 Eastman Kodak 181 1/8-11 1/8-6.14% 27,800 General Electric % 136,300 General Motors 40-7 ½ 971,300 Proctor & Gamble 77 ¾ 66 1/2-11 ¼ % 13,800 US Steel ,300 E8. Suppose you are studying the effects of poverty and plan to construct a data set whose cases are the villages in Bolivia. Name some variables that you might study.

9 Section 2: Discrimination in the Workplace: Inference Overall, the exploratory work on the Westvaco dataset in Section 1 shows that older workers were more likely than younger workers to be laid off and were laid off earlier. One of the main arguments in the court case was about what the patterns in the data mean: Can you infer from the patterns that Westvaco has some explaining to do, or are they the sort of patterns that tend to happen even in the absence of discrimination? A comprehensive analysis of Martin v. Westvaco will have to wait until the end of the course. For now, you can get a pretty good idea of how the analysis goes by working with a subset of the data. The ages of the ten hourly workers involved in Round 2 of the layoffs, arranged from youngest to oldest, were 25, 33, 35, 38, 48, 55, 55, 55, 56, and 64. The three workers who were laid off were ages 55, 55, and 64. Display 1.9 shows these data on a dotplot. Display 1.9 Hourly workers laid off retained pay = "H" To simplify the statistical analysis to come, it helps to condense the data into a single number, called a summary statistic. One possible summary statistic is the average, or mean, age of the three workers who lost their jobs = 58 years Knowing what to make of the data requires balancing two points of view. On one hand, the pattern in the data is pretty striking. Of the five workers under age 50, all kept their jobs. Of the five who were 55 or older, only two kept their jobs. On the other hand, the number of workers involved is small: only three out of ten. Should you take seriously a pattern involving so few cases? Imagine two people taking sides in an argument that was at the center of the statistical part of the Martin case.

10 Martin: Westvaco: Look at the pattern in the data. All three of the workers laid off were much older than the average of all workers. That s evidence of age discrimination. Not so fast! You re looking at only ten workers total, and only three positions were eliminated. Just one small change and the picture would be entirely different. For example, suppose it had been the 25-year-old instead of the 64- year-old who was laid off. Switch the 25 and the 64, and you get a totally different set of averages. (s in bold are those selected for layoff.) Actual Data: Altered Data: See! Make just one small change, and the average age of the three workers who were laid off is lower than the average age of the others. Average Laid Off Retained Actual Data Altered Data Martin: Westvaco: Martin: Not so fast yourself! Of all possible changes, you picked the one most favorable to your side. If you d switched one of the 55-year-olds who got laid off with the 55-year-old who kept his job, the averages wouldn t change at all. Why not compare what actually happened with all the possibilities? What do you mean? Start with the ten workers, and pick three at random. Do this over and over, to see what typically happens, and compare the actual data with the results. Then we ll find out how likely it is that the average age of those laid off would be 58 or greater. This dialogue between Martin and Westvaco describes one age-neutral method for choosing which workers to lay off: Pick three workers completely at random, with all sets of three having the same chance to be chosen.

11 Display 1.10 shows the results of 200 repetitions. Display 1.10 Results of 200 repetitions Mean 58 = 58 Out of 200 repetitions, only 12, or 6%, gave an average age of 58 or greater. So it is not at all likely that simply by chance you d pick workers as old as the three Westvaco picked. Did the company discriminate? There s no way to tell from the numbers alone Westvaco might have a good explanation. On the other hand, if our simulation had told us that an average of 58 or greater is easy to get by chance alone, then the data would provide no evidence of discrimination and Westvaco wouldn t need to explain. To better understand how this logic applies to Martin v. Westvaco, imagine a realistic argument between the advocates for each side:

12 Martin: Westvaco: Martin: Westvaco: Martin: Westvaco: Martin: Look at the pattern in the data. All three workers laid off were much older that average. So what? You could get a result like that just by chance. If chance alone can account for the pattern, there s no reason to ask us for any other explanation. Of course you could get this result by chance. The question is whether it s easy or hard to do. If it s easy to get an average as large as 58 by drawing at random, I ll agree that we can t rule out chance as one possible explanation. But if an average that large is really hard to get from random draws, we agree that it s not reasonable to say that chance alone accounts for the pattern. Right? Right. Here are the results of my simulation. If you look at the three hourly workers laid off in Round 2, the probability of getting an average age of 58 or greater by chance alone is only 6%. And if you do the same computations for the entire engineering department, the probability is a lot lower, about 1%. What do you say to that? Well I ll agree that it s really hard to get an average age that extreme simply by chance, but that by itself still doesn t prove discrimination. No, but I think it leaves you with some explaining to do! In the actual case, Martin and Westvaco reached a settlement out of court before the case went to trial. The logic you ve just seen is basic to all statistical inference, but it s not easy to understand. In fact, it took mathematicians centuries to come up with the ideas. It wasn t until the 1920s that a brilliant British biological scientist and mathematician, R.A. Fisher, realized that results of agricultural experiments may be analyzed in a way similar to the above to see whether observed differences could be attributed to chance alone or to treatment. This logic of using randomization as a basis for statistical inference will be used throughout the course.

13 Exercises E9. Revisit the idea of simulation. This time, consider all 14 hourly workers. Use as your summary statistic the number of hourly workers laid off who were 40 or older. The ages listed here are those of the hourly workers, with the ages of those laid off in bold. Note that, of the ten hourly workers laid off by Westvaco, seven were 40 or older a. Write the 14 ages on 14 slips of paper and draw 10 at random to be chosen for layoff. How many of the 10 are age 40 or older? b. The dot plot in Display 1.12 shows the results of 50 repetitions of this simulation. Estimate the probability that, by chance, seven or more of the ten workers who were laid off would be age 40 or older. Display Over40 c. Do you conclude that the proportion of laid-off workers age 40 or older could reasonably be due to chance alone, or should Westvaco be asked for an explanation? Why?

14 E10. The ages of the ten hourly workers left after Round 1 are given here. The ages of the four workers laid off in Rounds 2 and 3 are shown in bold. Their average age is a. Describe how to simulate the chance of getting an average age of or more. b. Perform your simulation once and compute the average age of the four hourly workers laid off. c. The dotplot in Display 1.13 shows the results of 200 repetitions of this simulation. What is your estimate of the probability of getting an average age as great or greater than Westvaco did if four workers are picked at random for layoff in Rounds 2 and 3 from the ten hourly workers remaining after Round 1? Display Average = d. What is your conclusion?