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Stats4us
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Exercises |
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Stats4us
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Exercises |
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Tutorial sessions may at the student's request include practice at analysing data from research projects, or answering typical applied statistics exam questions and set exercises.
This page provides a small sample of the style of practice exam or exercise questions, data analysis and applications which Joan Osborne has prepared and compiled for student learning purposes (tutoring and short courses), primarily in the field of Biostatistics (e.g. Quantitative Biology). Many of the exercises below refer to MINITAB printouts, but are also available for students using SPSS.
IMPORTANT NOTE: The questions below are designed for educational and instructional purposes only. With the exceptions of Question 7 (see acknowledgement) and Question 11, the practical research studies and data presented are entirely fictional and of Joan's invention. - They simulate hypothetical 'real world' cases, but are NOT based on actual studies and results. They have NO research validity.
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| 18 EXAMPLE QUESTIONS: | Q.1 | Q.4 | Q.7 | Q.10 | Q.13 | Q.16 | |
| (on this page) | Q.2 | Q.5 | Q.8 | Q.11 | Q.14 | Q.17 | |
| Q.3 | Q.6 | Q.9 | Q.12 | Q.15 | Q.18 |
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Pogona vitticeps ( photo from: http://www.angelfire.com/pq/beardeddragon/Photos ) The following data (see Table) give the abundance over four seasons of the Inland Bearded Dragon Pogona vitticeps. This species of bearded dragon lives in semi-arid woodland and desert shrubland of the eastern interior of Australia. It is omnivorous, feeding on insects, smaller lizards, vegetable matter and flowers. It is a "sun-loving" species and breeding follows good winter rains and associated abundant food supplies, followed by warmer spring months. The data were collected over a 12-month period, with a 10-day trapping period in each season. The Table provides the number of individuals trapped in each season. To determine male and female individuals, scale markings were used. Juveniles were identified by a shorter body length (snout to vent). Test the null hypothesis that the ratio of the numbers of males, females and juveniles remains constant over all four seasons (use a = 0.05). You must include the alternative hypothesis, and comment on the outcome from the statistical testing. |
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TABLE: Numbers of Bearded Dragon Pogona vitticeps over seasons; male female and juvenile individuals.
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Freshwater crab, Austrothelphusa transversa ( photo from: http://www.amonline.net.au/sand/news/freshwater_crabs.htm) The data in the table below come from a study of water loss in the fresh water crab, Austrothelphusa transversa. This species is widespread in the arid and semi-arid areas of Australia. To escape the dry weather, these crabs construct burrows in clay soils which they sometimes seal with mud. The burrows are usually around 60 cm long and are found in areas which experience annual flooding. The humid air trapped inside the burrows gives the crabs enough moisture to survive until wet weather returns. Following heavy rain, the crabs often appear in large numbers in the claypans of river floodplains. For this study the crabs were weighed and then kept at different relative humidities, and weighed again after six days of starvation. Weight loss in milligrams was computed for each crab.
The purpose of the statistical analysis of these data is to establish whether a linear relationship exists between relative humidity and weight loss. (a) Plot these data points on the provided graph paper. Do not join the points. (b) Refer to the MINITAB printout below. Write out the regression equation and then accurately draw this regression line onto the graph paper.
(c) Is the null hypothesis Ho: b = 0 rejected, testing at a = 0.05? Using information from the MINITAB printout below, elaborate on this testing procedure. (d) Formulate a conclusion for the study.
The following MINITAB printout was obtained:
MTB > PRINT C1 C2 Name c1 = 'WTmg' c2 = 'PERC' ROW WTmg PERC 1 8.98 0.0 2 8.14 12.0 3 6.67 29.5 4 6.08 43.0 5 5.90 53.0 6 5.83 62.5 7 4.69 75.5 8 4.20 85.0 9 3.72 93.0
MTB > MEAN C1 MEAN = 6.0233 MTB > MEAN C2 MEAN = 50.389 MTB > DESCRIBE C1 C2 N MEAN MEDIAN TRMEAN STDEV SEMEAN WTMG 9 6.023 5.900 6.023 1.736 0.579 PERC 9 50.4 53.0 50.4 32.2 10.7
MIN MAX Q1 Q3 WTMG 3.720 8.980 4.445 7.405 PERC 0.0 93.0 20.8 80.3
MTB > NOBRIEF MTB > REGRESS C1 1 C2
The regression equation is WTmg = 8.70 - 0.0532 PERC
Predictor Coef Stdev t-ratio P Constant 8.7036 0.1916 45.44 <0.001 PERC -0.053192 0.003256 -16.34 <0.001
s = 0.2967 R-sq = 97.4% R-sq(adj) = 97.1%
Analysis of Variance SOURCE DF SS MS F P Regression 1 23.488 23.488 266.89 <0.001 Error 7 0.616 0.088 Total 8 24.104
Obs. PERC WTMG Fit Stdev.Fit Residual St.Resid 1 0.0 8.9800 8.7036 0.1916 0.2764 1.22 2 12.0 8.1400 8.0653 0.1594 0.0747 0.30 3 29.5 6.6700 7.1345 0.1200 -0.4645 -1.71 4 43.0 6.0800 6.4164 0.1018 -0.3364 -1.21 5 53.0 5.9000 5.8844 0.0992 0.0156 0.06 6 62.5 5.8300 5.3791 0.1065 0.4509 1.63 7 75.5 4.6900 4.6876 0.1283 0.0024 0.01 8 85.0 4.2000 4.1823 0.1499 0.0177 0.07 9 93.0 3.7200 3.7568 0.1704 -0.0368 -0.15
MTB > CORR C1 C2 Correlation of WTmg and PERC = -0.987 |
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Zebra Finch (male and female) ( photo from: http://www.birdsinbackyards.net ) Birds exhibit adaptations which may be related to the type of food they eat. Birds from different taxonomic families, and even birds within the same family, exhibit different food preferences. For a University third-year Biology project, a student studied the zebra finch, Taeniopygia guttata. The student was to determine whether birds fed preferentially on particular seed types. For one recording period, six (6) individuals were placed in an enclosure and observed for a 90-minute period. Six seed types, each in excess of all the birds’ requirements, were placed randomly in separate trays in the enclosure. The frequencies with which the birds chose different seed types were recorded, and are listed below:
(a) State appropriate hypotheses and then, at a = 0.05, test whether seed preferences exist for the zebra finch. (b) Complete the appropriate statistical test. (c) Formulate a conclusion for this preliminary investigation. (d) For additional marks, complete appropriate subdivisions on the analysed data. Comment on the findings. |
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Muja Coal Mine at Collie ( photo from: http://www.griffincoal.com.au/Mining.html ) A collaborative research team of microbiologists and geochemists investigated the potential for sulphate-reducing bacteria to mediate acid mine drainage in the Collie (WA) coal basin. One laboratory experiment was conducted to test the effects of pH on the growth of one strain of sulphate-reducing bacteria. The amount of bacterial growth was assessed visually and the culture tubes were ranked from: 1 to 7 least growth most growth
(a) Using the MOST APPROPRIATE method to establish whether there was a correlation between pH and bacteria growth (test at a = 0.05), list the null and alternative hypotheses. (b) Formulate a conclusion for this experiment. |
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The spread of 'die-back disease' through jarrah forest ( photo from: http://www.naturebase.net/content/view/902/963 ) The spread of Phytophthora cinnamomi infection, commonly known as 'die-back' or 'Phytophthora root-rot disease', is having a devastating impact on native plant species in the southwest of Western Australia. In Eucalyptus species the symptoms of the infection resemble those of plant water stress. Increased leaf diffusive resistance is a characteristic symptom of a water-stressed plant. An increase in leaf diffusive resistance indicates that stomatal pores are closing, in an attempt to conserve water. Little is known about the effect of P. cinnamomi on hybrid Eucalyptus species. A plant physiologist obtained sixteen (16) E. marginata x E. patens seedlings and decided to investigate the effect of P. cinnamomi inoculation on these plants. The plant physiologist also planned to investigate whether increased lime levels in the soil led to reduced P. cinnamomi infection of these hybrids. Thus, the design for this investigation consisted of two independent variables: FACTOR A: P. cinnamomi inoculum condition, with two levels of this variable. FACTOR B: soil lime condition, with two levels of this variable. Each treatment condition was replicated FOUR (4) times. The dependent variable was leaf diffusive resistance, measured in seconds per square centimetre (s cm-2). The results from this investigation are recorded below:
A two-way ANOVA with replication was used to analyse these data, and a summary table of this analysis is provided below, with a tabulation of the main effects and interaction means.
(a) Formulate null and alternative hypotheses for testing for normality and homogeneity of variances. Using the first of the MINITAB printouts below, indicate acceptance or rejection at a = 0.05. Be sure you indicate what information from the printout gave acceptance or rejection of H0. (b) Complete all appropriate null and alternative hypotheses for testing for differences between means (Factor A and factor B) and the interaction effect, then indicate acceptance or rejection at a = 0.05 using the second of the MINITAB printouts below. Be sure you indicate what information from the printout gave acceptance or rejection of H0. (c) Provide a plot of the ANOVA interaction between the two FACTORS (A and B) – graph paper is provided.
(d) Using information from the analysis of variance summary, the interaction plot and relevant mean s cm-2, interpret these results. (e) Scheffé tests are NOT required. Explain why such testing is not completed here?
MINITAB printout for Phytophthora cinnamomi trial follows :
Test for Equal Variances: s/cm-2 versus INOCULUM, LIME
95% Bonferroni confidence intervals for standard deviations
INOCULUM LIME N Lower StDev Upper 0 0 4 2.12409 4.3112 25.8539 0 2 4 2.48536 5.0445 30.2511 1 0 4 5.31092 10.7795 64.6432 1 2 4 4.06190 8.2443 49.4404
Bartlett's Test (normal distribution) Test statistic = 2.75, p-value = 0.433
Levene's Test (any continuous distribution) Test statistic = 1.67, p-value = 0.226
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MTB > ANOVA 'S/CM' = INOCULUM ! LIME; SUBC> Means INOCULUM ! LIME.
Analysis of Variance (Balanced Designs)
Factor Type Levels Values INOCULUM fixed 2 0 1 LIME fixed 2 0 2
Analysis of Variance for s/cm-2
Source DF SS MS F P INOCULUM 1 5094.4 5094.4 89.30 <0.001 LIME 1 3240.5 3240.5 56.80 <0.001 INOCULUM*LIME 1 2822.3 2822.3 49.47 <0.001 Error 12 684.6 57.0 Total 15 11841.7
MEANS
o = no inoculum (i.e. no P.
cinnamomi), INOCULUM N s/cm-2 0 8 21.350 1 8 57.037
o = no lime, LIME N s/cm-2 0 8 53.425 2 8 24.962
Interaction Means INOCULUM LIME N s/cm-2 0 0 4 22.300 0 2 4 20.400 1 0 4 84.550 1 2 4 29.525
MTB > Table 'P. cinn' 'Lime'; SUBC> Means 's/cm-2'; SUBC> N 's/cm-2'.
Tabulated Statistics ROWS: P. cinn COLUMNS: Lime
0 2 ALL
0 22.300 20.400 21.350 4 4 8
1 84.550 29.525 57.038 4 4 8
ALL 53.425 24.962 39.194 8 8 16 CELL CONTENTS -- s/cm-2:MEAN N |
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( photo from: http://www.co.rockland.ny.us/health/FCH/hhc.htm )
In a study of the length of time spent on individual home visits by public health nurses, data were reported on length of home visit, in minutes, by a sample of 80 nurses (see Table below). A record was made of each nurse’s age and the type of illness of each patient visited. The researchers wished to obtain from their investigation answers to the following questions, using a two-way ANOVA at a = 0.05,
TABLE: Length of home visit in minutes by Public Health Nurses of different age groups,for four different Patient Types.
A two-way ANOVA with replication was used to analyse these data, and provided below are the summary table of this analysis, plus the table of means, and the interaction means plot.
(a) Testing for Homogeneity of Variance was completed for this data set, prior to any ANOVA analysis. Use the information below to determine whether the null hypothesis of variance equality is accepted or rejected. State the test statistic and indicate the basis of acceptance or rejection of the null hypothesis.
(b) Now complete all appropriate null and alternative hypotheses for testing of main effects difference between means and interaction, and indicate acceptance or rejection at a = 0.05 (see below) – stating how you arrived at these conclusions.
(c) Complete Scheffé (a = 0.05) testing where appropriate. You should show the two tables associated with each set of Scheffé calculations, the first set will show “calculations”, the second table will appropriately tabulate means providing underscoring and (or) like letters.
(d) Using information from the analysis of variance summary, the Scheffé testing, and the interaction graph, interpret the results.
Tabulated Statistics ROWS: AGE COLUMNS: ILLNESS
Cardiac Cancer HIV Tuberculosis ALL
20-29 23.000 35.200 33.200 20.000 27.850 5 5 5 5 20
30-39 25.600 34.000 34.400 23.400 29.350 5 5 5 5 20
40-49 26.200 39.800 40.200 25.600 32.950 5 5 5 5 20
50 and 29.200 48.000 46.400 28.800 38.100 over 5 5 5 5 20
ALL 26.000 39.250 38.550 24.450 32.063 20 20 20 20 80
CELL CONTENTS -- TIME (minutes):MEAN N
Worksheet was saved on 26/3/2006 General Linear Model
Factor Levels Values AGE 4 20 30 40 50 ILLNESS 4 1 2 3 4
Analysis of Variance for TIME (minutes)
Source DF Seq SS Adj SS Adj MS F P AGE 3 1246.84 1246.84 415.61 26.24 < 0.001 ILLNESS 3 3769.04 3769.04 1256.35 79.33 < 0.001 AGE*ILLNESS 9 211.21 211.21 23.47 1.48 0.174 Error (Within) 64 1013.60 1013.60 15.84 Total 79 6240.69
Unusual Observations for TIME
Obs. TIME Fit Stdev.Fit Residual St.Resid 7 45.0000 35.2000 1.7797 9.8000 2.75R 29 26.0000 34.0000 1.7797 -8.0000 -2.25R 66 40.0000 48.0000 1.7797 -8.0000 -2.25R 70 60.0000 48.0000 1.7797 12.0000 3.37R 74 55.0000 46.4000 1.7797 8.6000 2.42R
R denotes an obs. with a large st. resid. Means for TIME
AGE Mean Stdev 20-29 27.85 0.8899 30-39 29.35 0.8899 40-49 32.95 0.8899 50 and over 38.10 0.8899 ILLNESS 1 cardiac 26.00 0.8899 2 cancer 39.25 0.8899 3 HIV 38.55 0.8899 4 tuberculosis 24.45 0.8899 AGE*ILLNESS 20 1 23.00 1.7797 20 2 35.20 1.7797 20 3 33.20 1.7797 20 4 20.00 1.7797 30 1 25.60 1.7797 30 2 34.00 1.7797 30 3 34.40 1.7797 30 4 23.40 1.7797 40 1 26.20 1.7797 40 2 39.80 1.7797 40 3 40.20 1.7797 40 4 25.60 1.7797 50 1 29.20 1.7797 50 2 48.00 1.7797 50 3 46.40 1.7797 50 4 28.80 1.7797
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FIGURE 1: Two interaction representations for the four different age groups and four different Patient Types - Length of home visit in minutes by Public Health Nurses.
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MTB > %Vartest 'TIME' 'AGE' 'ILLNESS'; SUBC> Confidence 95.0. Executing from file: Vartest.MAC
Macro is running ... please wait
Homogeneity of Variance
Response TIME Factors AGE ILLNESS ConfLvl 95.0000
Bonferroni confidence intervals for standard deviations
Lower Sigma Upper n Factor Levels
1.39486 2.91548 17.2742 5 20 1 2.93760 6.14003 36.3797 5 20 2 2.06892 4.32435 25.6218 5 20 3 0.33830 0.70711 4.1896 5 20 4 0.80057 1.67332 9.9144 5 30 1 2.16620 4.52769 26.8266 5 30 2 2.01852 4.21900 24.9976 5 30 3 1.60828 3.36155 19.9172 5 30 4 1.28377 2.68328 15.8985 5 40 1 1.19129 2.48998 14.7531 5 40 2 1.70500 3.56371 21.1150 5 40 3 1.29266 2.70185 16.0085 5 40 4 1.14225 2.38747 14.1458 5 50 1 3.62791 7.58288 44.9286 5 50 2 2.92198 6.10737 36.1862 5 50 3 0.62380 1.30384 7.7253 5 50 4
Bartlett's Test (normal distribution) Test Statistic: 31.488 P value : 0.008
Levene's Test (any continuous distribution) Test Statistic: 0.962 P value : 0.504 |
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Tour de France ( photo from: http://www.velonews.com/images/int/5740.7135.f.jpg )
Acknowledgement - The following text and data (excluding the exercises 'a' and 'b') are extracted from "STATISTICS" by R.S. & J.S.Witte (Sixth Edition, Copyright © 2001 Harcourt, Inc., ISBN 0470001828; pages 308, 326 & 337; and are used here by permission of John Wiley & Sons, Inc., Hoboken, New Jersey.
During the 1998 Tour de France, the world’s best-known bicycle race, some cyclists were expelled for attempting to enhance their performance by 'blood-doping' with the synthetic hormone erythroprotein, or EPO, which stimulates the production of oxygen-bearing (and fatigue-inhibiting) red blood cells. An investigator wants to determine whether this type of blood-doping increases the endurance of athletes under controlled laboratory conditions. (We’ll ignore the very real medical, ethical and legal issues raised by this type of experimentation.) Volunteer athletes from the local track team are randomly assigned to one of two groups: a blood-doped group, which receives a prescribed amount of EPO, and a non-blood-doped group, which receives a comparable amount of a harmless neutral substance. Subsequently, after an appropriate interval of time has elapsed, each athlete runs on a rapid treadmill until exhausted. Total time on the treadmill is used as the measure of endurance. Preliminary studies of blood doping with EPO reveal an unexpected phenomenon: Lightweight athletes in both the blood-doped and non-blood–doped groups have better endurance scores than heavier athletes. This factor complicates the search for the effect of blood-doping on endurance scores. In subsequent experiments, therefore, it is advantageous to pair athletes with similar body weights. Before collecting the data, the athletes are matched for body weight, beginning with the two lightest and ending with the two heaviest. Once a member of a given pair has been randomly assigned to either the blood-doped or the non-blood-doped group, the other member of that pair is automatically assigned to the remaining group. Thereafter, the athletes are treated in the same way as in the original blood-doping experiment; that is, endurance scores are obtained both for athletes who are blood-doped with EPO and for athletes who receive a harmless neutral substance. The data presented in the Table below shows an Endurance Score (minutes) for the athlete subjects who have been paired on the basis of weight and randomly allocated to the groups.
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TABLE: Endurance Scores (minutes) for the athlete subjects running on a rapid tread mill.
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(a) Carry out the MOST SUITABLE statistical test (using a = 0.05) to determine whether the synthetic hormone erythroprotein (EPO) has caused any change in the endurance of athletes. (b) Summarise the findings of this experiment. |
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Tailings discharge into storage cell. ( photo from: http://www.tailings.info/deposition.htm ) At a tailings storage facility in the Goldfields region of Western Australia, an underground nickel mine was depositing tailings into a new cell of the storage. The tailings monitoring bores on the western edge of the facility showed a range of readings, at the January 2006 sampling, indicating movement perhaps of materials from the storage into the ground water. Previous readings from all bores had all been in the 25,000 – 35,000 mg/L range. Hypersaline water was being used in the nickel extraction process with the residual slurry deposited into a tailings cell. The five bores had six readings taken from each bore, and the TDS (total dissolved solids) in mg/L are given below in the Table below. The bore locations are depicted on Figure 1 below. TABLE: TDS in mg/L from five Tailings Dam Monitoring Bores (TDMB) at the nickel mine, January 2006 sampling.
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FIGURE 1: Location of the tailings monitoring bores, at the nickel mine. Statistical analyses are required to compare mean TDS (mg/L) of the five bores. i) List the dependent and the independent variables. ii) Formulate the appropriate null and alternative hypotheses for comparison between bores of mean TDS, and you will indicate acceptance or rejection at a = 0.05. iii) Complete an analysis of variance (ANOVA) on these data, testing at a = 0.05, to compare mean TDS (mg/L) at the monitoring bores (TDMB). Complete the analysis by hand showing all workings then check using MINITAB or SPSS. Provide your MINITAB or SPPS hard copy printout, with appropriate labelling and description (LIS and MTW files [or SPPS files] will be on disc). iv) If a difference between means exists (at a = 0.05) use Scheffé tests to elucidate where any difference between means are statistically significant. You will manually complete the Scheffé tests, and provide the required two tables associated with the calculations: One table gives calculations, the second table presents means. v) Finally, formulate a written summary of this sampling using the information from the analysis of variance (ANOVA) summary, with any explanations you may see as being biologically / environmentally feasible. |
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'Yabby', Cherax destructor ( photo from: http://www.deh.gov.au/cgi-bin/species-bank/sbank-treatment2.pl?id=69119 )
An aquaculture research unit was invited to complete a study; one aspect was to consider growth (as measured by gm of edible flesh) of the introduced yabby Cherax destructor (Family Parastacidae), under different conditions.
Farmers in the York to Kelleberrin area of W.A. were approached, with five farms having appropriate dams for the stock – it was a requirement that each farm had four discrete dams which were “similar” (e.g. drainage, catchment, construction type, dam sealing structure). However, it appeared likely that conditions (e.g. rainfall, soil type) would vary between farms over this 50-100 km radius, thus each LOCATION (farm) was considered a BLOCK in the experimental design.
There was a different experimental set-up in each dam, with the same set-ups on all farms. The experimental set-ups are listed below: · Control, no shelter, no shade and no “salt” addition (“salt” has been documented as on acting as pathogens in the waters). · Shelter – here poly-piping has been inserted (the poly-piping has been suggested as providing shelter for the crustaceans, and also a “ready made” facility for protection). · Shelter + shade (shade trees planted around the dam were suggested as a environmental amelioration with perhaps lower water temperatures). · Shelter + shade + “salt” addition (“salt” has been document as acting as pathogens in the water). In the study, weight of only one yabby from each location was recorded (statisticians had checked this was a valid procedure by sub-sampling a number of animals, and releasing them back into the dam). Data are present in the Table below. It is noted that the experimental design implemented is a randomised block layout, with the five farm (LOCATIONS) blocks, and there are four experimental treatments which have been randomly allocated to the four dams on each farm. |
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TABLE: Dry weight in gm, of yabbies (Cherax destructor; Family Parastacidae) under experimental set-ups in dams on wheat-belt W.A. farms. Experimental Set-up (within One Dam)
i) List all the dependent and independent variables. ii) Complete the appropriate analysis of variance on the data testing at a = 0.05. The analysis of variance must be completed manually showing all workings. Show the critical values from the Fcritical table (at a = 0.05) used to ascertain significance. All appropriate hypotheses will have been presented and then acceptance or rejection is to be stated. iii) You will have checked your calculations with MINITAB, and you will provide the MINITAB or SPSS printout. iv) Use the Scheffé test (a = 0.05), where appropriate, to compare mean dry weights for the experimental treatments. You will manually complete the Scheffé tests, and provide the required two summary tables for any set of Scheffé testing. Hypotheses will have been formulated, presented, and their acceptance or rejection given. v) You are also to Scheffé test (a = 0.05) the “BLOCK” means, and establish where there are location differences. You will manually complete the Scheffé tests, and provide the required two summary tables for any set of Scheffé testing. Hypotheses will have been formulated, presented, and their acceptance or rejection given. vi) Provide a written summary and conclusion for this research trial. |
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Bluefin tuna. ( photo from: http://members.tripod.com/gill_fisher/fsblue.jpg )
Southern Bluefin Tuna, Thunnus maccoyii, are being used for an exploratory aquaculture venture off the southern coast of Australia. The researchers are asked to show the relative growth curve of Southern Bluefin Tuna to a State Fisheries Department for comment.
The relative growth curve of fishes is best represented using a log10\ log10 plot – both the horizontal and vertical axes are of log10 scale. Prepare the following two plots for presenting to the Fisheries Department, using either 4 cycle by 2 cycle, or 2 cycle by 2 cycle scales :
· Total length (Y1) against standard length (X), and · Body length (Y2) against standard length (X)
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TABLE 1: Southern Bluefin Tuna, Thunnus maccoyii, data for relative growth curves, length (cm) n = 9.
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Pseudomys sp. ( photo from : http://www.zoo.latrobe.edu.au/Staff/pbf/marsupials.htm )
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The following small mammal species and numbers of individuals were trapped on a control site of Spinifex grasslands with acacias, and on a rehabilitated iron-ore waste-dump site at Newman, Western Australia.
Animals were caught in pit-fall traps which had been dug into the ground, with 100 traps per site. Trapping was at regular monthly intervals over two years. The control site was 3 km north of the waste dump rehabilitation site. i) Calculate the Shannon Index of Diversity (H), and J the Evenness or Equitability Index, for the Control Site and the Rehabilitation. ii) Use Sorenson’s Index of Similarity (ISs) to compare the two sites. iii) Represent these data graphically in a way that clearly shows differences between the two trap locations (the Control Site and the Rehabilitation), and species differences. iv) Using both the species listing (see Table) and the calculated diversity studies, comment on these data. TABLE : Species and numbers of small mammals trapped on the control and the rehabilitation, Newman 1996-1998, recaptures are excluded (- not captured).
Source of data: From original paper jointly authored by Joan Osborne. |
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Hippocampus sp. ( photo from: http://www.divegallery.com/seahorse_page3.htm )
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Seahorses are generally found in shallow waters where human disturbance is intensive and extensive. Seahorse habitats – seagrass meadows, coral reefs and mangroves – are among the most threatened in the world. Additionally, seahorses are overexploited, as pre-packaged medicines and for the aquarium and curio trade, because demand exceeds supply. The common species, Hippocampus subelongatus, has been found to successfully grow and reproduce in captivity. Both growth rates and reproductive success have been influenced by diet. Long-term studies have identified the calanoid copepod Gladioferens imparipes Thomson to be a potential live food source for Hippocampus subelongatus. As an adjunct to investigating seahorse growth in captivity and thus fish farming potential – environmental chemists investigated the value of the calanoid copepod Gladioferens imparipes as a food source. When fed on different algal diets, Gladioferens imparipes produces different unsaturated fatty acids (HUFA’s) and these have the potential to affect growth rate of the seahorses. Copepods raised on different algal diets were fed to seahorses and the growth rate of the seahorses was established. Seahorse growth rate was the taken as the length (mm) of the seahorse after six months. Juvenile seahorses, at the commencement of the study were same length (ANOVA testing on lengths, and the Levene Test on variances). As part of a collaborative research program extending to Asia and co-ordinated from America, the study was repeated in five different laboratories where there had been records of successful ornamental fish farming in captivity. Two laboratories were located in Western Australia, one in Tasmania, one in Malaysia, and one in Chicago (Illinois USA). In each laboratory copepods were raised on the following diets, and then fed to seahorses. The conventional bakers yeast diet was used as a control comparison. Length (mm) of the seahorse was recorded after six months.
In this study the length weight of only one seahorse, from each laboratory for one diet, has been recorded (statisticians had checked this was a valid procedure by sub-sampling a number of seahorse and releasing them back into the aquaria). Data are presented in the Table below. The experimental design implemented is a randomised block layout, with the five laboratories (LOCATIONS) blocks and there are five experimental treatments, which have been randomly allocated to the five aquaria in each laboratory.
TABLE: Length in mm, of seahorses (the common species Hippocampus subelongatus) under experimental aquaria set-ups in different laboratories.
i) List all the dependent and independent variables. ii) Complete the appropriate analysis of variance on these data testing at a = 0.05. The analysis of variance must be completed manually showing all workings. Show the critical values from the Fcritical table (at a = 0.05) used to ascertain significance. All appropriate hypotheses will have been presented and then acceptance or rejection is to be stated. iii) Provide manually completed Scheffé tests for both Factor A and Factor B (a = 0.05). Include workings, and the two standard Tables presented for each of Factor A and Factor B. iv) Using GLM check your ANOVA calculations. Also complete on MINITAB Tukey’s Testing for both Factor A and Factor B (Blocking). Provide a copy of the MINITAB or SPSS printout. Ensure you include appropriate annotation on the printout to identify the study, and indicating what the Levels of Factor A and Factor B refer to. v) For the written summary for this research trial you are to use the ANOVA and the Tukey’s Testing data (a = 0.05) – so there is now the requirement to produce two Tukey’s Test summary tables. One summary table will relate to Factor A – the comparison of mean (n=5) length in mm of seahorses, from the different treatments (diets), where means are rank ordered and like means are underscored or given the same letter. The second summary table will consider the “BLOCK” means, and establish where there are location differences (Tukey’s Test a = 0.05). vi) Finally, prepare a written summary and conclusion for this research trial. |
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For Background Interest only ( text from: http://www.divegallery.com/seahorse_page3.htm ) : The profile view in the above photo of Hippocampus sp. makes gender identification easy - females don't have a brood pouch for carrying the developing young (this one is female). For reproduction, the female deposits a long string of eggs into the male's brood pouch, where the developing brood is nurtured until the baby seahorses emerge, looking like miniatures of their parents, commonly 6-12 millimeters in length. Once born, the young seahorses receive no care from their parents, and are particularly vulnerable to predators. The species in the photo (Hippocampus kuda) is commonly known as the spotted seahorse, yellow seahorse, or estuary seahorse. It is characteristically smoother in appearance than most other species, of medium size, 7-17 cm. |
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( photo from: http://albanymarronfarm.com.au ) An aquatic ecologist was studying the invertebrate fauna of Lake Inspiration, a shallow freshwater lake of southern and inland Western Australia, with physio-chemical attributes having been deemed conducive to intensive crustacean ("yabby" or "marron") farming. At this stage in the studies the ecologist was specifically addressing the abundance and seasonality of the invertebrate fauna (as a potential food source for the farmed crustaceans). The data on the following page provide a listing of the summer and winter fauna of this lake (see Table 1) with frequencies. Sampling was from a 10 cm3 volume. i) Is this a quantitative or qualitative frequency distribution (and just very briefly explain why)? ii) Represent these data graphically in a way that clearly shows the differences between species (keeping the Phylum, Class or Order listing as given), and the differences over time. TABLE 1: Abundance (volume 10 cm3) of invertebrate fauna from a fresh water lake insouthern Western Australia, over seasons - frequencies for species is given.
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( photo from: http://www.uow.edu.au/science/frontiers/2003/ ) An environmental scientist was investigating the mercury concentration of a river that had residues from a factory discharged into it. The scientist was required to determine if there were changes in mercury concentration downstream from the factory outlet. Five (5) sampling stations were chosen and six readings were made at each station. The data below are mercury (Hg) concentrations (mg/ml) at the five sampling stations which were situated every 500 metres downstream of the outlet. Station 1 Station 2 Station 3 factory outlet 500 m downstream 1,000 m downstream 56 46 41 54 42 44 60 44 46 63 49 40 60 44 39 57 40 36
Station 4 Station 5 1,500 m downstream 2,000 m downstream 39 28 41 33 38 36 37 34 43 29 42 31 i) List the dependent and the independent variables. ii) Formulate the appropriate null and alternative hypotheses for the environmental study, and indicate acceptance or rejection at a = 0.05.
iii) Complete an analysis of variance (ANOVA) on these data, testing at a = 0.05, to compare mean Hg levels at the sampling stations. Complete the analysis by hand showing all workings, then check using MINITAB or SPSS. Provide your MINITAB or SPPS printout, with appropriate labelling and description.
iv) If a difference between means exists (at a = 0.05) use Scheffé tests to elucidate where any differences between means are statistically significant. Manually complete the Scheffé tests, and provide the required two tables associated with the calculations, one table gives calculations, the second table presents means.
v) Finally, formulate a written summary of this sampling using the information from the analysis of variance (ANOVA) summary, with any explanations you may see as being biologically / environmentally feasible. |
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Eucalyptus globulus seedlings. ( photo from: http://www.forestalnet.com/tempo2000/fotos2000.htm ) As part of a collaborative research program between Universities in Chile (South America) and W.A., a plant species trial was established near Lake Rapel in central Chile where rainfall averages 560 mm and the dry season is long (7-9 months). Four fertiliser amendments (which included a control - no amendment) were applied to Eucalyptus globulus (Tasmanian blue-gum) seed which had been sown at the beginning of winter. Four months after fertiliser application seedling survival numbers and the height of the seedlings were recorded. Heights are given for statistical analysis. The experimental design implemented for this study was a randomised block lay-out with five block replicates of each of the fertiliser amendments (see Table 1). At this field study site there had appeared to be a trend for soil 'conditions' to improve in the more southerly blocks. In the absence of different treatments, it was anticipated growth would be better in the southern blocks. Thus fertiliser amendments were randomly allocated to each block. TABLE 1: Randomised block lay-out for five replicates of the four fertiliser amendments on a study site having a possible soil fertility trend in the direction indicated.
After four months the height of seedlings (cm) (one randomly selected seedling per treatment / block combination) was recorded and these data are given in Table 2. The objective of the trial was to establish whether fertiliser addition had improved plant growth. Comparisons between fertiliser amendments were required, and block variability with respect to plant growth, was to be investigated.
TABLE 2: Height (cm) after four months of Eucalyptus globulus (Tasmanian blue-gum) seedlings, grown using one of four different fertiliser amendments.
i) List all the dependent and independent variables. ii) Complete the appropriate analysis of variance on the data testing at a = 0.05. The analysis of variance must be completed manually showing all workings. Show the critical values from the Fcritical table (at a = 0.05) used to ascertain significance. All appropriate hypotheses will have been presented and then acceptance or rejection are to be stated. Check your calculations with MINITAB, and provide a MINITAB or SPSS printout. iii) Would you recommend Scheffé testing for block differences? Elaborate. iv) Now use the Scheffé test (a = 0.05), where appropriate, to compare mean heights of Eucalyptus globulus seedlings. Manually complete the Scheffé tests, and provide the required two summary tables for any set of Scheffé testing. Hypotheses will have been formulated, presented, and their acceptance or rejection given. v) Provide a written summary and conclusion for this research trial. |
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School of sharks ( photo from: http://www.aloha.com/~lifeguards/sharintr.html )
A zoologist was studying the physiology of two species of sharks (Family Carcharhinidae) known from the waters off Rottnest Island, Western Australia. One minor aspect of this study was to look at the phosphorus content of different body organs. The phosphorus content (measured in mg/100 g) of each of four different organs was determined in two species, and the data are tabulated below.
Complete the appropriate ANOVA for these data, ensuring all relevant means are compared.
i) List all dependent and independent variables. ii) Use MINITAB (or SPPS) to complete the appropriate analysis of variance. Show the critical values from the Fcritical table (at a = 0.05) used to ascertain the significance of all main effects, and the interaction. All appropriate hypotheses will have been presented and then acceptance or rejection are to be stated.
iii) Provide an appropriate summary of means either via MINITAB, SPSS or another package (e.g. Excel), or manually.
iv) Complete Scheffé (a = 0.05) testing where appropriate. You should show two tables associated with each set of Scheffé calculations, the first set will show your 'calculations' , the second table will appropriately tabulate means.
v) Summarise and interpret the data set. - An interaction graph (presented as a Figure) is to be included and referred to during the interpretation. |
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A zoologist was studying the physiology of two species of sharks (Family Carcharhinidae) known from the waters off Rottnest Island, Western Australia. One minor aspect of this study was to look at the phosphorus content of different body organs. The phosphorus content (measured in mg/100g) of each of four different organs was determined in two species, and data are tabulated below. Thus the design of this investigation considered two independent variables. FACTOR A: Body Organ, with four levels of this variable. FACTOR B: Species, with two levels of this variable.
Each treatment condition was replicated FOUR (4) times. The dependent variable was phosphorus content in mg/100g. The results from this investigation are recorded below:
A two-way ANOVA with replication was used to analyse these data. The summary table of this analysis and the table of means are provided below.
(a) Testing for Homogeneity of Variance was completed for this data set, prior to any ANOVA analysis. Use the information below to determine whether the null hypothesis of variance equality is accepted or rejected. State the test statistic and indicate the basis of acceptance or rejection of the null hypothesis. (b) Complete all appropriate null and alternative hypotheses and indicate acceptance or rejection at a = 0.05 stating how you arrived at these conclusions.
(c) Provide an interaction graphical summary (see below for table of means).
(d) Complete Scheffé (a = 0.05) testing where appropriate. You should show two tables associated with each set of Scheffé calculations: the first will show your 'calculations', and the second will appropriately tabulate means providing underscoring and (or) like letters.
(e) Interpret the results, using information from the analysis of variance summary, the Scheffé testing, and the interaction graph.
MTB > Table 'SPECIES' 'ORGAN'; SUBC> Means 'Pmg/100g'; SUBC> N 'Pmg/100g'. Tabulated Statistics
ROWS: SPECIES COLUMNS: ORGAN
Heart Gills Liver Kidney ALL
Dusky 88.82 106.25 210.20 186.40 147.92 4 4 4 4 16
Long-nosed 79.35 98.08 196.93 176.23 137.64 4 4 4 4 16
ALL 84.09 102.16 203.56 181.31 142.78 8 8 8 8 32
CELL CONTENTS -- Pmg/100g:MEAN N
MTB > ANOVA 'Pmg/100g' = SPECIES! ORGAN; SUBC> Means SPECIES! ORGAN.
Analysis of Variance (Balanced Designs)
Factor Type Levels Values SPECIES fixed 2 1 2 ORGAN fixed 4 1 2 3 4
Analysis of Variance for P mg/100g
Source DF SS MS F P SPECIES 1 845 845 99.76 <0.001 ORGAN 3 82191 27397 3236.02 <0.001 SPECIES*ORGAN 3 28 9 1.11 0.366 Error 24 203 8 Total 31 83267
MEANS
SPECIES N P mg/100g | ||||||||||||||||||||||||||||||||||||||||||||||
