Lake Pollution

(Adapted from the Project Intermath ILAP Lake Pollution)

Background

The Great Lakes provide the main water supply for many people in the United States and Canada. Additionally, they are commercially fished, provide transportation, and are a source of recreational facilities. Unfortunately, a considerable amount of waste and sewage is dumped into these lakes. This waste includes a massive amount of phosphates which have been traced to detergents, insecticides, and other chemicals such as DDT and mercury. Extensive pollution kills off the fish, as well as other forms of animal and plant life.

The pollution of the Great Lakes has, in the past few decades, been a significant issue on the agenda of both local and regional environmentalist groups. There have been a number of clean-up initiatives that have been started in the past 20 years which have greatly improved the outlook for the future of this natural waterway. The primary polluters have been some of the heavy industries that use the lakes as a means of transportation to get their products to both domestic and international markets. Some of the most significant violators are the steel industries around the Pittsburgh area on Lake Erie and around the Gary, Indiana area of Lake Michigan, and the mining industries around the Superior/Duluth areas of Lake Superior. Both of these industries have made significant strides in recent years to clean up their operations and their efforts are starting to slowly pay off.

You have been hired as an analyst by a regional aluminum processing plant that is located on the shores of Lake Erie. Two years ago, the plant, in cooperation with several other major polluters on the Great Lakes, implemented a fairly comprehensive clean-up plan which temporarily calmed the fears of concerned environmentalists. Recently, however, a study was published in an environmentally friendly journal which implied that the clean-up plans that had been implemented by the major polluters were not effective. The journal article basically claimed that the clean-up plans were mere eyewash to quiet the environmental watchdog groups. This revelation has energized all of the key players involved in the Great Lakes pollution issue and each side is currently assessing their ability to defend their position in a court of law.

Your boss has put you in charge of the team which will study this issue. She has given you two years worth of data, and she wants some quality statistical analysis which will withstand any scrutiny by the environmentalists. The data consists of weekly test samples that were taken from Lake Erie and analyzed for various attributes (so a total of 104 samples for the two years).

Data

The data can also be downloaded directly into your calculator from the file LakePollution.8lg, which is located in the Statistics folder (in the Dr. Mellor folder) on the computers in the lab. You will need to use the Graph Link program, which is installed on the three computers in the rear of the lab which have signs saying "Graph Link". The six columns in the table contain the following data:

week: The week in which the sample was collected.

alum_ppm: The amount of aluminum pollutants in the weekly test sample in parts per million (ppm).

oth_ppm: The amount of other pollutants in the weekly test sample in parts per million (ppm).

puretime: The amount of time that it takes to purify a weekly test sample in seconds.

epacode: This variable is 0 if the sample is not "polluted" per EPA guidelines, and 1 if the sample is "polluted" per EPA guidelines.

temp: The temperature of the sample in degrees Fahrenheit.

week	alum_ppm	oth_ppm	puretime	epacode	temp
1	36.2846	59.739	12.105	1	70.0692
2	35.1909	43.6978	37.864	0	68.8021
3	38.5753	45.4508	67.275	1	71.9412
4	29.8268	55.6974	45.286	1	62.2027
5	32.7359	43.6578	81.583	0	64.9527
6	33.4072	56.6556	6.297	1	66.1605
7	31.5494	49.9503	32.174	1	63.9796
8	29.483	59.2501	15.345	1	63.8191
9	28.3125	59.2778	103.952	1	61.8358
10	33.3508	30.8085	44.684	0	67.2914
11	35.4065	50.986	16.497	1	68.2996
12	40.8595	59.0104	28.704	1	72.8905
13	38.0167	57.1471	67.667	1	70.5572
14	48.5467	48.4696	36.098	1	80.3929
15	38.2178	47.4461	17.372	1	70.1867
16	35.7263	37.0215	33.567	0	68.3591
17	44.3164	52.0141	23.307	1	75.7433
18	33.7658	53.8839	109.182	1	66.486
19	38.6447	35.8155	1.579	0	70.8557
20	44.4616	43.8406	2.717	1	76.2052
21	37.8688	58.4949	72.884	1	70.2122
22	33.2331	56.73	131.37	1	66.7332
23	39.3477	45.6032	57.226	1	72.0411
24	34.2471	37.7449	58.106	0	66.5119
25	38.9567	45.796	16.483	1	71.3661
26	37.2145	44.0806	11.4	1	69.1802
27	39.3465	55.0693	63.58	1	71.9098
28	34.671	31.5343	8.304	0	67.6962
29	26.6301	57.3854	18.576	1	60.7372
30	30.8516	30.2706	91.128	0	64.3912
31	30.8723	42.1799	5.037	0	63.3019
32	29.7076	44.9115	17.931	0	62.7968
33	42.7976	40.9325	38.233	1	74.7648
34	44.4128	57.4171	52.001	1	75.9615
35	28.4938	30.2881	12.81	0	62.3881
36	32.8682	46.1233	30.378	0	66.9608
37	35.1048	39.4139	17.783	0	67.8019
38	40.9571	46.7199	0.608	1	73.3128
39	28.7226	36.3994	27.31	0	62.4025
40	32.7272	42.1318	65.823	0	65.277
41	30.2132	54.0219	18.391	1	64.411
42	29.0897	34.4747	56.934	0	62.9142
43	34.0419	49.2455	168.731	1	67.592
44	28.1496	43.044	29.275	0	62.4702
45	35.2347	58.2225	5.884	1	67.3556
46	37.2364	57.4492	4.351	1	70.5495
47	29.162	53.2136	42.727	1	63.2839
48	40.446	30.933	32.152	0	72.5938
49	35.7201	48.8363	11.04	1	69.567
50	29.2285	32.6732	137.548	0	62.9715
51	42.9233	53.6791	40.148	1	75.5686
52	34.9017	31.7362	69.898	0	68.4558
53	26.6193	35.7099	21.848	0	59.771
54	32.8131	39.6966	19.566	0	64.9707
55	30.609	48.3676	25.052	0	63.1579
56	28.5616	42.1876	1.559	0	61.6976
57	30.4919	53.2427	19.179	1	63.7523
58	39.593	47.2459	44.036	1	71.4984
59	32.4821	31.4635	46.615	0	65.7153
60	35.1442	43.3965	22.226	0	68.2051
61	33.953	44.8305	18.446	0	67.0155
62	29.4919	46.8731	150.477	0	63.7843
63	28.4642	39.5781	15.591	0	60.9638
64	26.1369	37.0135	14.602	0	60.3454
65	35.1675	37.8624	4.265	0	67.6119
66	37.4473	37.1184	40.328	0	69.8011
67	37.6916	36.5494	33.648	0	69.9018
68	24.6438	41.2847	2.789	0	58.6964
69	30.0035	48.9843	91.2	0	62.4671
70	26.0824	50.6743	146.261	0	59.0033
71	31.5738	48.537	33.855	1	64.0707
72	33.683	33.1112	43.153	0	65.7709
73	30.7659	54.359	26.966	1	64.6611
74	28.5338	56.7471	192.736	1	61.7126
75	29.9502	34.9079	58.938	0	63.9489
76	39.0303	43.4007	7.306	1	71.5458
77	40.7136	34.6556	29.328	0	72.5465
78	47.5382	42.0995	42.509	1	80.2774
79	27.8366	55.4403	15.455	1	60.49
80	34.5073	42.9701	13.94	0	67.0364
81	33.2366	56.2188	18.492	1	66.2645
82	38.185	39.4222	15.752	0	71.2584
83	31.3074	33.5046	6.902	0	63.958
84	32.5488	38.6973	22.425	0	66.5546
85	30.3595	59.6348	25.319	1	63.0884
86	36.6709	38.3582	75.05	0	70.0097
87	37.7908	46.5821	3.351	1	70.9777
88	33.5284	36.829	9.954	0	66.3875
89	33.988	35.3832	28.603	0	67.7514
90	30.681	56.9938	41.422	1	63.1022
91	33.2926	33.039	9.068	0	66.6784
92	34.0935	37.8167	6.904	0	66.2453
93	37.59	51.4286	13.276	1	70.7118
94	35.6707	37.3356	0.195	0	67.7272
95	35.0608	30.3771	23.171	0	68.2565
96	30.6652	37.7568	40.732	0	64.8418
97	33.3158	43.9597	78.473	0	67.0687
98	32.9712	50.1189	2.845	1	66.6186
99	43.8148	47.6849	28.864	1	76.5815
100	37.1492	53.0739	53.365	1	68.9791
101	34.0851	34.636	13.298	0	66.2452
102	37.8289	36.0122	106.067	0	71.2823
103	40.5501	31.8986	228.922	0	73.7126
104	41.6061	49.6269	57.271	1	74.019

Part 1

Your first job is to "see" what the data looks like, to get a better feel for it. Consider the three variables alum_ppm, oth_ppm and puretime.

A. For each of the three variables, construct a histogram and stemplot. Be sure to think carefully about the intervals for the histograms, and how much to round off for the stemplots.

B. Discuss the shape and symmetry of the various plots in A. Do any of the distributions look normal?

C. Do a time plot for each of the three variables. Are there any noticeable patterns?

D. Produce the five-number summary and associated boxplot for each variable.

E. Compute the mean and standard deviation for each variable.

Part 2

It occurs to you that there might be a relationship between the temperature of the sample and the amount of pollution detected, and you decide to look more carefully into this question.

A. Plot the variable alum_ppm against the temperature (temp). Does there appear to be a relationship? What kind?

B. Find the equation of the line which best fits the plot. What is r, the correlation? What is r²? What do these numbers tell you about the linear model? Give a physical interpretation for the slope of your line?

C. What amount of aluminum pollutants would you expect to find in a weekly sample with a temperature of 65 degrees Fahrenheit?

D. Repeat A and B for oth_ppm. Do aluminum pollutants and other pollutants have similar relationships to temperature?

Part 3

So far, you've ignored the variable epacode. You need to look at this to see what is meant by "polluted."

A. Your first problem is to determine what standard the EPA used to decide whether a sample was "polluted." Your boss has only given you the relevant EPA guidelines - all 5,000 pages - and, in a typical snafu, the index was not included. You decide it's easier just to figure it out from the data. Find the total number of pollutants in each sample (the sum of the aluminum and other pollutants), and compare these to the epacode. Explain why it is reasonable to assume that the EPA considers a sample "polluted" when the total pollutants is 80 ppm.

B. What percentage of the samples were polluted?

Since you are working for the aluminum company, your boss wants you to take a hard look at the aluminum pollutants and see what kind of data you can collect to help the company's case.

C. You recall the average amount of other pollutants (oth_ppm) from your calculations in Part 1. Assuming this is the amount of other pollutants in your sample, what level of aluminum pollutants will cause the EPA to label the sample polluted?

D. Assuming the amount of aluminum pollutants (alum_ppm) has a normal distribution with the mean and standard deviation you found in Part 1, what is the probability that the amount of aluminum pollutants will be at or above the level you determined in question C in a single sample?

E. On the other hand, you decide it might be better to consider the average amount of aluminum pollutants. What is the probability that the average value of alum_ppm over a two-year period (104 samples) will be at or above the level you determined in question C?

F. You remember that the measurement of aluminum pollutants was highly dependent on the temperature. Recall the value of alum_ppm you would expect if the sample temperature were 65 degrees Fahrenheit, from Part 2. What would the level of other pollutants have to be for the sample to be polluted?

G. Remember that the level of other pollutants (oth_ppm) was not dependent on temperature, so you can assume it has the same distribution at 65 degrees as in your sample data. What is the probability that the average value of oth_ppm over a two-year period would reach the level in F? What theorem did you use to compute this, since the distribution for oth_ppm is not normal?

Part 4

Your boss wants a "solid" estimate of the amount of aluminum pollutants currently in the Great Lakes.

A. Provide an estimate for the population mean of the amount of aluminum pollutants in a weekly test sample.

B. Using the z-statistic, provide an interval estimate of the population mean of the amount of aluminum pollutants in a weekly test sample. Use a confidence level of 90%. Explain how you can use the z-statistic even though you do not know the population standard deviation (just the sample standard deviation you computed in Part 1), and why this will give you a reasonable answer.

C. Interpret your answer to the previous question in terms of probability.

Another method in your arsenal is hypothesis testing, and you decide to apply this to the aluminum pollutants. You will continue to use z-procedures.

D. You want to show that the true population mean of the amount of aluminum pollutants in a weekly test sample is less than the amount you computed in Part 3C. Test this claim at a significance level of 0.05. Use a sketch to support your conclusion.

E. Historical data over the years shows that the true mean of the amount of aluminum pollutants that are in a weekly test sample is 35 ppm. Perform a Type II error analysis on your test from D assuming that the historical data is accurate. Use a sketch to support your conclusions.

F. What is the power of your test in E? What can you do to increase the power of your test without increasing the probability of other errors? What is the possible drawback with using this method?

G. (Extra Credit) Since we did not know the population standard deviation exactly, it would have been more precise to use t-procedures than z-procedures; do so. You will have to work out for yourself how to do the error analysis in question D using the t-statistic!

Part 5

A. Looking back over your previous reports, your boss mentions that you never gave a confidence interval for the linear models you computed in Part 2. Compute these confidence intervals, and test each linear model against the hypothesis of no linear relationship.

Your boss has now received all of your reports to date, with the details of your computations. She wants you to prepare a summary of your results, conclusions and recommendations for her to present to the Board of Directors. Your summary should be approximately two pages long, written in coherent paragraphs and correct English. Remember that you are giving your recommendations as to how the company should present the results in court to their best advantage; the Board is not interested in your opinions of how the company conducts its business.

However, your conscience is pricking you for helping the polluters of the Great Lakes, some of the continent's greatest natural resources. So you have decided to also write up the opposing case and leak it to the environmentalists (anonymously, of course - you don't want to lose your job!). Again, this should be a summary of approximately two pages of your results and how the environmentalist lobby could present them to greatest effect.