Appendix 5¾Mid-term and Final Exams

First mid-term exam:

Please enter your last four digits here _________ and turn this exam in with your answers

E538¾Exam 1¾February 16, 2000¾D. Parkhurst ©2003, D.F. Parkhurst

Note before proceeding:

· Please write the LAST FOUR DIGITS of your student number, but no other identifying information, on every page you hand in.

· Start each question on a new page so I can separate pages to grade them.

· Please number all your pages in the upper right-hand corner.

· At least once in each problem, label your main answer with its correct units (mm, grams, etc.), if it has units. (This will be worth points.)

· Keep at least 4 significant digits in all calculations and at least 3 in all answers. (Remember that leading zeroes are not “significant digits,” so 27.13, 0.2713, and 0.002713 all have 4 significant digits.) Following this instruction will also gain you points.

· If you use probability tables anywhere and don’t find the exact values you need, just pick the nearest available value.

· If you present your work in detail, it will be easier for me to give you partial credit for any work that might be only partly correct.

· If you recognize that some value you calculate doesn’t make sense, I may give you some credit if you explain why you think the value is wrong.

· Remember to make plots or rough sketches wherever it would help you to understand a problem, or to make your answers clearers.

· If any of your answers requires assumptions not already given here, be sure to state those assumptions.

· Please write clearly. If I can’t read your answers, I may ask you later to type them before I grade them!

This exam has two parts. Part I is CLOSED BOOK, and Part II is OPEN BOOK. To help you budget your time, it’s a good idea to read through both parts before beginning Part I, but YOU MUST HAND IN PART I BEFORE YOU OPEN ANY BOOKS FOR PART II. Please write your answers on 8.5” by 11” paper.

I. CLOSED BOOK PART

For this part you may not use any books, notes, or calculators. Please place all such items on the floor until you have handed in this part.

1. (8 points) Explain concisely but precisely why “least-squares” regression has that name.

2. (8 points) What is the 68-95-99.7 rule?

3. (16 points) An ecologist uses a stratified random sampling scheme to place forty quadrats, each 4 m square, in a large forest stand. In each quadrat, she measures the total light (L), received between March 1 and October 31, the total number (N) of tree seedlings less than 2 m tall, the number (W) of white oak seedlings, and the number (T) of tulip poplar seedlings.

A. For which pair of variables, L and N, or W and T, would regression be more appropriate, and for which pair would correlation be more appropriate? Explain your choice.

B. In the regression case, which would be the dependent (response) variable and which the independent (explanatory) one? Explain.

II. OPEN BOOK PART

For this part you may use any books and notes you have with you. You may also use a pocket-sized calculator (but not a portable computer). However, because several students do not have regression functions in their calculators, use of such functions is off limits for this exam.

4. (20 points) You believe that some quantity of interest to you is distributed approximately normally, with a mean of 95 and a variance of 20. What fraction of the items in that distribution have values: A. larger than 82? B. larger than 102? C. Between 95 and 102? D. If you took a very large sample from that distribution, what would you predict that its IQR would be?

5. (20 points) A state air-pollution agency performed numerous unannounced spot checks at power plants one year. In 45% of the cases, sulfur emissions were excessive (in violation) and in 32% of cases, particle emissions were too high. Both violations occurred simultaneously in 22% of cases. Determine: A. In what percentage of cases was there a sulfur violation or a particle violation? B. what percentage violated for sulfur only? C. what percentage of those cases with sulfur violations also had particle violations. D. Also determine from the mathematical definition whether or not the two types of violation occurred independently or not.

6. (15 points) Suppose (the numbers are fictitious) that during the 1990’s it rained[1] on 25% of all days in New York City and it rained on 30% of all days in Boston. Further suppose that it rained in Boston on 45% of the days that it rained in New York.

Let us assume as a first approximation that those same probabilities and dependencies will hold from 2000–2009. Then in this latter period, what fraction (or percentage) of days when it rains in Boston would we expect to see rain in New York?

Show your calculations in detail.

7. (20 points) Devise and describe an experiment to study some phenomenon of interest to you. It should involve a single factor that you control at two or more levels, and a single response variable. It should also involve randomization and blocking. Be sure to describe the factor, the levels, the response, the randomization, and the blocking. Also explain why the randomization and the blocking that you propose will make your experiment more valuable.

J Your experiment should not involve ski wax, shoe soles, nor nutrients added to water.

Second mid-term exam:

Please enter your last four digits here _________ and turn this exam in with your answers

®Nothing in this exam¾neither calculations nor interpretations¾should be Bayesian¬

E538¾Exam 2 ¾March 30, 2000¾D. Parkhurst

Note before proceeding:

· Please write the LAST FOUR DIGITS of your student number, but no other identifying information, on every page you hand in.

· Start each question on a new page so I can separate pages to grade them.

· Please number all your pages in the upper right-hand corner.

· At least once in each problem, label your main answer with its correct units (mm, grams, etc.), if it has units. (This will be worth points.)

· Keep at least 4 significant digits in all calculations and at least 3 in all answers. (Remember that leading zeroes are not “significant digits,” so 27.13, 0.2713, and 0.002713 all have 4 significant digits.) Following this instruction will also gain you points.

· If you use probability tables anywhere and don’t find the exact values you need, just pick the nearest available value.

· If you present your work in detail, it will be easier for me to give you partial credit for any work that might be only partly correct.

· If you recognize that some value you calculate doesn’t make sense, I may give you some credit if you explain why you think the value is wrong.

· Remember to make plots or rough sketches wherever it would help you to understand a problem, or to make your answers clearers.

· If any of your answers requires assumptions not already given here, be sure to state those assumptions.

· Please write clearly. If I can’t read your answers, I may ask you later to type them before I grade them!

This exam has two parts. Part I is CLOSED BOOK, and Part II is OPEN BOOK. To help you budget your time, it’s a good idea to read through both parts before beginning Part I, but YOU MUST HAND IN PART I BEFORE YOU OPEN ANY BOOKS FOR PART II. Please write your answers on 8.5” by 11” paper.

I. CLOSED BOOK PART

For this part you may not use any books, notes, or calculators. Please place all such items on the floor until you have handed in this part.

1. (10 points) Suppose that five years ago, students in a SPEA class placed collector plates on the SPEA building roof, and measured the amount of soot and dust collected in ten eight-hour overnight periods. Another group has recently repeated that study during similar weather conditions to see whether the heating plant has changed its emissions. (They recognize that some of the particles collected may come from other sources, so the study is not definitive.) You obtain the data, and test (with alpha set to 10 percent) versus .

Suppose you obtain a P value of 0.17, with this year's mean being about 20% lower than the 1996 mean. An IDS reporter interviews you to write an article about your study for that newspaper. She asks what your P value means. How would you answer?

2. (12 points) The North Fork of Salt Creek is one of the major streams flowing into Monroe Reservoir, Bloomington's water supply. Suppose, hypothetically, that during a particular one‑hour period in June, a volume of 6 x 10⁵ L of water discharged into the reservoir via this stream. In the same period, 30,000 cysts of the intestinal parasite Giardia lamblia were carried in with that water. Assume that the cysts were distributed randomly and independently throughout the hour's incoming water volume. (For your post‑exam consideration, is independence really likely here?)

During that hour, researchers obtained one 100 L sample of the incoming water. They filtered the sample, added a stain, and then inspected a subsample consisting of one fifth of the filtrate under fluorescent light in a microscope. Now, because of your superior statistical knowledge, the other scientists on the team are asking you what theoretical sampling distribution would be appropriate for describing each of the following:

A. The probability distribution for the number C₁ of cysts in the original 100 L sample.

B. Assuming that there were C₁ cysts in the original sample, the probability distribution for the number of cysts C₂ that will be in the subsample.

Give:
(i) the name of each distribution you choose, and
(ii) why you chose each. (The same theoretical distribution may or may not apply to both situations.)
(iii) Also provide the values of all necessary parameters of these two distributions, stating them as precisely as you can. You do not have to give formulas for individual probabilities, however.

II. OPEN BOOK PART

For this part you may use any books and notes you have with you. You may also use a pocket-sized calculator (but not a portable computer). However, because several students do not have regression functions in their calculators, use of such functions is off limits for this exam.

3. (20 points) A microbial ecologist is interested in the effects of heavy metals on soil fungi. She obtained several hundred samples from a series of control sites that have low, background levels of metals, and found that these contained an average of 5,000 meters of fungal hyphae (root‑like structures) per gram of soil. The corresponding standard deviation was about 1,200 m, and the data appeared to be fairly symmetrically distributed when dot‑plotted.

In a follow‑up study, the ecologist plans to take 20 soil samples from each of the control sites, and to calculate the mean for each site of hyphal length per gram of soil. If there is no serious spatial autocorrelation among the samples at each site, then she can treat the observations at each as independent. If they were independent, and if fungal growth conditions are similar between the old and new sampling periods, what would be the approximate sampling distribution she can expect to obtain for these site‑wise sample means? Express your answer in two ways:

A. As a written description of the probability density function for the new means. Be as explicit as you can, and provide numerical values for any parameters involved.

B. With a sketch of the PDF (probability density function) of the distribution. Show the mean and standard deviation of the distribution on the sketch. (You need not put numbers on the vertical density axis.)

4. (20 points) Calculate a set of 95% (two‑sided) confidence limits for the true mean of the population from which the following seven numbers were sampled: 3.2, 4.8, 1.3, 3.7, 2.7, 2.9, 3.0 [seconds]. Summarize your results in a sentence; your summary should involve the concept of probability. If you must make any assumptions, state them.

5. (45 points) A state legislature has passed a law authorizing the state Air Pollution Control Board (APCB) to establish and enforce regulations regarding industrial air pollution. The new statute also stipulates that the APCB should be comprised of one industrial representative, one environmental professional, and one private citizen; all are appointed by the governor.

The industrial representative appointed is a statistician who has cunningly proposed that the Board's staff can impose fines on a particular industry only if staff can prove “using scientific standards" that the industry has exceeded a regulatory limit for a given pollutant. Specifically, if the staff suspects an infraction, they are to sample the industry's discharges two times a month, at random, for three months. Then, they are to perform a standard hypothesis test at the a = 5% level to decide whether the mean discharge exceeds the regulatory standard for the particular pollutant ("Standard" here means a test based on an appropriate probability distribution, NOT a randomization test.) The other two board members like the sophisticated sound of this proposal, and approve it as law.

The first time this regulation is exercised, the staff believes that the Zylol Corporation is emitting more than the allowed 100 mg m^-3 of xylene from its stack. They take the required random samples, and obtain these results from the lab: 111.4, 114.1, 107.9, 99.5, 97.3, and 106.0 mg xylene m^-3 air. The mean of these six concentrations is mg m^-3, and their variance is 43.26267.

A. Under the law described, should Zylol Corp. be fined? (Provide a P value for the data as part of your interpretation. The t table provided at the end of the exam should help you with that.)

B. If that t table were not available, what SPSS calculation would yield the P value, based on your t statistic. Be as specific as possible.

C. If the first reading had been 144.7 (about 33 mg m^-3 higher than 111.4), and the third had been 94.6 (roughly 13 mg m^-3 lower than 107.9), the mean discharge would have increased by about 3.3 mg m^-3 to a new value of , and the variance would have changed to 348.6387. (The six emission measurements are now 144.7, 114.1, 94.6, 99.5, 97.3, and 106.0 mg m^-3.) Could the company be fined under the law in this case? (Again, provide a P value.) Discuss why or why not.

D. Suppose you were a statistician for the local chapter of the Clean Air Defense Fund. Describe a reasonable alternative regulation (from your organization's point of view) that you might have proposed during the public hearings held before the version described above was approved.

6. (30 points) Because emissions from radioactive materials occur at random and independently, the number of decay events per any fixed unit of time is a random variable with a Poisson distribution. (Remember that with discrete distributions like the Poisson, you work by summing actual probabilities, rather than by integrating probability densities as you do when working with continuous distributions.)

Suppose you work in a lab, and a single sample of some hard-to-obtain material is brought in for counting in a beta counter. Twelve counts are observed in a 24-hour period, i.e., k=12. Your boss uses that datum to perform a hypothesis test of (counts per 24 hr) versus , with alpha set at 5 percent.

A. Using the Poisson probabilities given in the table below, what observed count would be required to reject that null hypothesis? That is, what is the critical value of k for this test?

B. Is the observed k=12 a "statistically significant" result in this case?

C. What is the definition of power as it applies to that test?

D. What would be the numerical value of the power of your boss's test if the true m were really 14 counts per 24 hr?

The values in the table below should be useful ¾ entries in the table are probabilities of observing various counts k when the true mean is either 10 or 14, and when a Poisson distribution holds. For each mean, the column to the left shows the probabilities for specific counts, while the column to the right shows cumulative probabilities. Sketching a distribution is likely to be helpful, too.

Poisson table for problem 6:

	10		14
k	P(K=k) (specific)	P(K³k) (cumulative)	P(K=k) (specific)	P(K³k) (cumulative)
0	0.000045	0.000045	0.000001	0.000001
1	0.000454	0.000499	0.000012	0.000012
2	0.002270	0.002769	0.000081	0.000094
3	0.007567	0.010336	0.000380	0.000474
4	0.018917	0.029253	0.001331	0.001805
5	0.037833	0.067086	0.003727	0.005532
6	0.063055	0.130141	0.008696	0.014228
7	0.090079	0.220221	0.017392	0.031620
8	0.112599	0.332820	0.030436	0.062055
9	0.125110	0.457930	0.047344	0.109399
10	0.125110	0.583040	0.066282	0.175681
11	0.113736	0.696776	0.084359	0.260040
12	0.094780	0.791556	0.098418	0.358458
13	0.072908	0.864464	0.105989	0.464448
14	0.052077	0.916542	0.105989	0.570437
15	0.034718	0.951260	0.098923	0.669360
16	0.021699	0.972958	0.086558	0.755918
17	0.012764	0.985722	0.071283	0.827201
18	0.007091	0.992813	0.055442	0.882643
19	0.003732	0.996546	0.040852	0.923495
20	0.001866	0.998412	0.028597	0.952092
21	0.000889	0.999300	0.019064	0.971156
22	0.000404	0.999704	0.012132	0.983288
23	0.000176	0.999880	0.007385	0.990672
24	0.000073	0.999953	0.004308	0.994980
…

Final exam:

Please enter your last four digits here _________ and turn this exam in with your answers

E538¾Exam 3¾April 30, 2001¾D. Parkhurst

Note before proceeding:

· Please write the LAST FOUR DIGITS of your student number, but no other identifying information, on every page you hand in.

· Start each question on a new page so I can separate pages to grade them.

· Please number all your pages in the upper right-hand corner.

· At least once in each problem, label your main answer with its correct units (mm, grams, etc.), if it has units. (This will be worth points.)

· Keep at least 4 significant digits in all calculations and at least 3 in all answers. (Remember that leading zeroes are not “significant digits,” so 27.13, 0.2713, and 0.002713 all have 4 significant digits.) Following this instruction will also gain you points.

· If you use probability tables anywhere and don’t find the exact values you need, just pick the nearest available value.

· If you present your work in detail, it will be easier for me to give you partial credit for any work that might be only partly correct.

· If you recognize that some value you calculate doesn’t make sense, I may give you some credit if you explain why you think the value is wrong.

· Remember to make plots or rough sketches wherever it would help you to understand a problem, or to make your answers clearers.

· If any of your answers requires assumptions not already given here, be sure to state those assumptions.

· Please write clearly. If I can’t read your answers, I may ask you later to type them before I grade them!

This exam has two parts. Part I is CLOSED BOOK, and Part II is OPEN BOOK. To help you budget your time, it’s a good idea to read through both parts before beginning Part I, but YOU MUST HAND IN PART I BEFORE YOU OPEN ANY BOOKS FOR PART II. Please write your answers on 8.5” by 11” paper.

I. CLOSED BOOK PART

For this part you may not use any books, notes, or calculators. Please place all such items on the floor until you have handed in this part.

1. (12 points) Suppose data are obtained for the weights [kg] of ninety adult male wildcats (thirty from each of three separate locations) and an analyst subjects those data to an analysis of variance to test for inequality of the means. Explain in words (as sentences, please) what the variances (or mean squares) in the numerator and denominator of the F ratio represent. You may provide equations if you wish, but if you do, be sure to explain them in words. (The best answers are likely to be short and to the point.)

2. (12 points) The manager of a small industrial wastewater treatment plant submits a report to the state water pollution control agency regarding the effects of the plant's effluents on the stream into which the effluents flow. One section of the report deals with dissolved oxygen concentrations in the stream above and below the discharge point. (High DO concentrations are desirable; low concentrations can lead to fish kills and other undesirable effects.) Data are provided for 12 monthly paired measurements of DO above and below that discharge point. There is a mean drop in DO concentration of 2.3 mg L^-1 (from 13.6 above the discharge to 11.3 below it), with the variance of the difference being 41.14.

The report then includes an analysis involving a significance test with the null hypothesis being versus the alternative (i.e. the alternative states that there is a reduction in mean DO as the stream flows by the plant). The resulting P value is 0.12, and the manager concludes that "any difference is just a result of random chance."

Explain why that conclusion is not a valid interpretation of the results just outlined.

3. (10 points) Before performing a t test for the difference between means of unpaired data, data analysts sometimes carry out what they call a "test for equality of variances." The hypotheses for this test are H₀: versus H₁: . Then, if the analysts reject that null hypothesis, they perform the type of unpaired t-test that we concentrated on the class, i.e. the type that does not assume equal variances. On the other hand, if the data don't lead to rejecting that null hypothesis, the analysts perform the "pooled- variance" test that is fully valid only if the two population variances really are equal. Explain why at least part of this decision-making process is flawed.

II. OPEN BOOK PART

For this part you may use any books and notes you have with you. You may also use a pocket-sized calculator (but not a portable computer). However, because several students do not have regression functions in their calculators, use of such functions is off limits for this exam.

4. (25 points) Suppose you are working with a group of limnologists who have data (hypothetical here) relating chlorophyll concentrations to phosphorus concentrations (C) in a group of six Minnesota lakes, as shown in the figure just below. One interesting result of this analysis is the slope estimate (b = 0.18 [mg chlorophyll] [mg P]^-1), which indicates the increase in chlorophyll concentration to be expected from a unit increase in C. Some scientists might consider it desirable to perform a significance test to determine whether (and, they might unfortunately think, "or not") the suggested increase was “real.” A one-tailed t test for versus yields the P value of 0.14 shown in the figure, so the result is "not statistically significant" at the conventional level.

Upon seeing that result, one of the limnologists exclaims “That’s odd. Usually increased phosphorus increases chlorophyll concentrations, but in these particular lakes, phosphorus concentration has no effect on chlorophyll concentration.” With your superior knowledge of statistics, you see that he is making the common error of treating the null hypothesis as true, merely because it couldn’t be rejected. So you suggest a reverse test to help in deciding whether these data provide evidence that any relationship between the two concentrations is negligible in a practical sense in this set of lakes.

You ask the limnologists to define what slope they would consider to be negligible, and they suggest a value of b = 0.25. That is, based on their professional judgment, they believe that if a unit increase in P concentration caused an increase of chlorophyll concentration of 0.25 mg m^-3 or less, that that would be biologically unimportant.

Your task is thus to test (at a = 0.05) the hypotheses H₀: Slope b = 0.25 versus H₁: Slope b < 0.25. (For this, you need to know that the standard error of the estimated slope, i.e., s_b, was provided by the regression software as 0.1442.) Use the t table on p. 6 of this exam to estimate the P value. Just use the nearest value from the table¾you don’t have to interpolate. Then write a few sentences describing your result and conclusions, to present to the limnologists.

0. (0 points) Please insert a sheet somewhere within your closed-book answers that has on it only the last four digits of your student number and a code name that I can use for posting your grade by email. (This will be a single list with everyone’s code names and corresponding grades.) This is optional¾if you would rather obtain your grade via the University's telephone system, you can omit this page. Please do NOT use any or all of your student number (that's against University rules), nor any code that would allow others to identify you easily.

5. (20 points) A state air pollution control agency receives regular monitoring data from a number of industries. One company, the XYZ Corporation, supplies the agency with daily data (5 days per week) on carbon monoxide emissions from their stack. The state finds that although the weekly means can vary substantially from week to week, the variance of daily values stays relatively constant through time, at a value of about kg² da^-2. You should use that estimated population variance in the analysis that follows, and ignore the sample variance.

The state has placed a limit of 1500 kilograms per day on the mean mass of carbon monoxide that XYZ is permitted to emit in any given week. That is, in any five-day work week, XYZ is allowed to emit kilograms. One week, the corporation provides the following data for its five work days: 997, 1591, 1348, 1806, and 1064 kg. The mean of those numbers is 1361.2, and because that is less than 1500 kg, there is no violation this week.

An agency staff member who is performing a general study of the effectiveness of the agency's regulations wants to estimate the probabilities that, based on those measurements, the true mean for this particular week is: A. less than 1450 kilograms per day, B. less than 1500 kilograms per day, and C. less than 1550 kilograms per day.

Noting that probabilities can be put on true means only by Bayesians, use Bayesian analysis with a non-informative prior to estimate those three probabilities.

(Problem 6 on next page)
6. (25 points) Consider again the smoking-lung cancer regression example from pages 194–197 of the notes. A few statistics from that example are gathered here:

n	12		10.17		1.756´10^-6
b0	6.912
b1	0.005496		526571.97		25.15

A. Determine the 95% confidence interval for the population intercept, b₀. The intercept is the true mean value on the line when .

B. Suppose you find, for one city that was not in the original dataset, that the smoking rate was 3800 cigarettes/person/year. Determine a lung-cancer death rate R [deaths/yr/10⁵ people] such that there is only a 10% chance that the true value would be higher than the R value you calculate for that city.

(For use in Problem 4.) A t table for n = 2–8 degrees of freedom. Values in the table are upper tail probabilities, i.e. , for the t values in the first column.

t	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8
0.0	0.500	0.500	0.500	0.500	0.500	0.500	0.500
0.1	0.465	0.463	0.463	0.462	0.462	0.462	0.461
0.2	0.430	0.427	0.426	0.425	0.424	0.424	0.423
0.3	0.396	0.392	0.390	0.388	0.387	0.386	0.386
0.4	0.364	0.358	0.355	0.353	0.352	0.351	0.350
0.5	0.333	0.326	0.322	0.319	0.317	0.316	0.315
0.6	0.305	0.295	0.290	0.287	0.285	0.284	0.283
0.7	0.278	0.267	0.261	0.258	0.255	0.253	0.252
0.8	0.254	0.241	0.234	0.230	0.227	0.225	0.223
0.9	0.232	0.217	0.210	0.205	0.201	0.199	0.197
1.0	0.211	0.196	0.187	0.182	0.178	0.175	0.173
1.1	0.193	0.176	0.167	0.161	0.157	0.154	0.152
1.2	0.177	0.158	0.148	0.142	0.138	0.135	0.132
1.3	0.162	0.142	0.132	0.125	0.121	0.117	0.115
1.4	0.148	0.128	0.117	0.110	0.106	0.102	0.100
1.5	0.136	0.115	0.104	0.097	0.092	0.089	0.086
1.6	0.125	0.104	0.092	0.085	0.080	0.077	0.074
1.7	0.116	0.094	0.082	0.075	0.070	0.066	0.064
1.8	0.107	0.085	0.073	0.066	0.061	0.057	0.055
1.9	0.099	0.077	0.065	0.058	0.053	0.050	0.047
2.0	0.092	0.070	0.058	0.051	0.046	0.043	0.040
2.1	0.085	0.063	0.052	0.045	0.040	0.037	0.034
2.2	0.079	0.058	0.046	0.040	0.035	0.032	0.029
2.3	0.074	0.052	0.041	0.035	0.031	0.027	0.025
2.4	0.069	0.048	0.037	0.031	0.027	0.024	0.022
2.5	0.065	0.044	0.033	0.027	0.023	0.020	0.018
2.6	0.061	0.040	0.030	0.024	0.020	0.018	0.016
2.7	0.057	0.037	0.027	0.021	0.018	0.015	0.014
2.8	0.054	0.034	0.024	0.019	0.016	0.013	0.012
2.9	0.051	0.031	0.022	0.017	0.014	0.011	0.010
3.0	0.048	0.029	0.020	0.015	0.012	0.010	0.009

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[1] We’ll use “rain” to indicate any kind of measurable precipitation for this question.