Biometry exams test 1
1. It has been speculated that butterfly wing morphology is used by birds to distinguish which individuals are palatable. An experimental biologist set out trays with 10 pinned butterfly individuals in her backyard to evaluate this claim. Five insects in each tray had Morphology A, the rest Morphology B. She then observed the tray from a blind and determined the first insect that a bird attempted to eat. After each trial she reinitialized her tray with intact insects. Eight birds were separately observed and Morphology A was the choice of 7 of them. Use your knowledge of statistics to decide if this constituted evidence that birds use such morphological cues. Do not concern yourself with any other prey characteristics of butterflies that might confound these observations.
2. Male fireflies are thought to utilize periodic pulses of bioluminescence to attract females. An animal behaviorist simulated these blinking lights and was able to attract an average of 6 fireflies a night over an extended research program. A skeptical colleague suggested that this had nothing to do with periodicity but simply was due to the general attraction of insects to lights. She left a light on continuously for a night and not a single firefly was attracted to it. The behaviorist argued his case, but she claimed that the attraction was the same and her result was a possible outcome even if the systems were equally attractive to fireflies. What would be the probability of this outcome based on her argument.
3. What is the power of a statistical procedure to distinguish between a normally distributed population with = 10, = 3 and an alternative hypothesized normally distributed population with = 11, = 3. Assume the design involves a sample size n = 10 and = 0.05.
4. Provide the shortest unbiased confidence intervals for and 2 based on the following sample.
10.1, 7.8, 8.4, 11.2, 5.6
5. Twenty postmenopausal women who had undergone a hysterectomy during their premenopausal years received synthetic estrogen therapy daily for four months. After treatment the following estrogen values were recorded:
61 58 54 54
81 56 81 75
61 80 92 59
63 83 71 58
82 92 69 94
Researchers would like to report upper and lower confidence intervals for the mean and variance of this category of women exposed to this treatment. Help them do this. Report shortest unbiased confidence intervals.
6. In a study of the effectiveness of an insecticide against fire ants a large area was sprayed. Past experience has shown that the average number of surviving colonies per experimental quadrant was 0.5 and that these colonies were randomly assorted. What is the probability of there being at least one colony surviving in the first quadrant examined?
7. Damselflies have 3 gills at the tip of their abdomen while larvae. The gills are often lost during intraspecific encounters. When 224 larvae were grouped at densities of 16 larvae/aquarium for 24 hours and then examined the following results were obtained:
# gills remaining on larvae # larvae
3 132
2 59
1 30
0 3
If we assume that the loss of any gill is independent of the loss of any other gill, calculate the expected number of larvae that should have been found in the above 4 categories.
8. A national testing firm has provided a standardized achievement test for biology students which has been shown around the country to result in an average score of 65 with a standard deviation of 5. I used this test in my course on 100 freshmen and they averaged 68.5. What can I conclude from this? Show work.
9. A researcher has determined through years of research that 85% of damselfly larvae which possess a complete complement of caudal lamellae will abandon the stalk they are perching on and swim away when approached by the hand of an observer. The researcher removes all 3 lamellae from 10 larvae and challenges these larvae in a similar experiment. One of these larvae abandons its perch and swims. Is there and evidence (at 95% confidence) that larvae modify their behavior based on the presence of lamellae?
10. I frequently have been driving my 1989 Honda Accord from Arlington to Austin, Texas for a number of years and have been keeping track of how many insects splatter on my windshield on typical Autumn days. On average 5 individuals are sacrificed to the great automotive god during each trip. I have just purchased a new Accord and plan a trip to Austin this weekend. What is the probability that my new car's windshield will be spared from the experience of an insect's splatter?
11. One way systematists can establish the amount of phylogenetically useful information that exists in their data set is by differentially weighting characters at random and generating phylogenetic trees from these new data sets. When the lengths of a number of such trees are examined they will be normally distributed if the characters contain no phylogenetic signal. A renowned odonate systematist randomly generated 1 million such theoretical trees and calculated a gl statistic equal to -0.693. He then had to decide whether to conclude that his data set had phylogenetic signal or not at a 95% confidence. Show him how to do this.
12. Summer water temperature released from a cooling pond of an electrical generating plant is mandated to be normally distributed with a mean of 30° C and a standard deviation of 1.5° C. If the temperature actually has a mean value of 32° C the utility must pay a substantial penalty. To demonstrate compliance, they take a total of 12 samples during the summer. Consider the mandated information to represent your null hypothesis and evaluate the power of this procedure to detect an actual mean temperature of 32° C. Assume the standard deviations of the null and alternative models to be equal.
13. Suppose it is know that in a certain large human population cranial length is normally distributed with a mean of 185.6 mm and a standard deviation of 12 mm. What is the probability that a random sample size 10 from this population will have a mean greater than 190.
14. Suppose it is known that 30% of a certain population are immune to some disease. If a random sample of size 10 is selected from this population, what is the probability that 2 or fewer of them will be immune.
15. Serum amylase determinations were made on a sample of 15 apparently healthy subjects. The sample yielded a mean of 96 units/100 ml and a standard deviation of 35 units/100 ml. Develop confidence limits relating to the parametric mean and variance. These should be the shortest unbiased intervals.
16. If one assumes a mean mutation rate per locus of 1x10 -6 and accepts McKusick's estimate of at least of at least 100,000 genes per person, then the probability that any individual will produce a mutation during his or her reproductive lifetime is 1x10 -6 x 1x10 5 = 0.1. I have been informed by a reliable source that a member of this class has produced 3 mutations during his/her reproductive life. Should we accept the null hypothesis of humanity for this subject or rather assert that this individual is an alien life form.
17. Mitochondria contain DNA and replicate independently from cell replication. A mutation occurs in mitochondrial DNA and results in 4 out of 10 mitochondria in a cell carrying a new message. Assume that cytokinesis (cell division) randomly separates the mitochondria such that 5 are contained within each descendant cell. What is the probability that all 4 mutant mitochondria wind up in the same descendant cell.
18. The average number of masked (due to diploidy) deleterious mutations per individual belonging to a continental population is 6.3. An individual belonging to the same species is obtained from an isolated population on an oceanic island. This individual is demonstrated to have only 1 deleterious masked mutation. Would you conclude that this individual was from a different population (in a statistical sense) from mainland individuals.
19. The length of the wings from 10 individuals belonging to the damselfly species Ischnura kellicotti in Arlington, Texas are (in mm): 22.9, 23.8, 25.0, 22.0, 23.0, 23.2, 24.1, 23.8, 23.0, 24.7, 24.5. Provide 95% confidence limits on the mean, variance, and unbiased coefficient of variation.
20. Published data indicate that the mean for I. kellicotti is 24.5 mm. Assess the correspondence of the Arlington data with this published information.
21. Ischnura kellicotti is a damselfly species whose life history is intimately tied to the water lily (Nuphar sp.). Larvae cling to these plants, use them as emergence platforms and hatch from eggs imbedded in the plant's tissue. Adults perch on the leaves, from where they attack nearby prey. It has been suggested that an individual leaf may constitute a territory which is defended by a resident damselfly individual. 1000 leaves were examined during a field study and the following number of I. kellicotti were observed per leaf. Do these data suggest that the joint occupancy of leaves by I kellicotti is less than expected based on the null hypothesis of random leaf occupancy by individuals.
# on leaf 0 1 2 more than 2
frequency 685 230 85 0
22. The above problem was reexamined by a more clever researcher. She suggested that leaf joint occupancy might be sex dependent. In other words, two individuals might be expected to occur jointly on a leaf more frequently than random if they were opposite sexes. To evaluate this hypothesis she considered the leaves jointly occupied and recorded the sexual identity of damselflies found:
15 with 2 males
20 with 2 females
50 with one male and one female
of the total number of damselflies observed, 250 were male and 150 were female. With that absolute frequency should we expect there to be each of the above three outcomes.
23. A budding damselfly taxonomist decided on revolutionizing dichotomous key presentation. Each couplet, according to his scheme, was going to represent a hypothesis. The first member of this couplet was H0 (the null hypothesis) and the second member H1. If the null hypothesis was rejected, users were advised to follow the directions associated with the alternative hypothesis. One couplet was to distinguish between two species known to have head widths averaging 4 and 5 mm, respectively. The standard deviation of head width was known to be 1 mm in each species. How often would an individual known to be a member of the second species (i.e., the second member of the couplet) be correctly identified as belonging to the species?
24. The dry weight of eggs dissected from Ischnura hastata are (in micrograms): 12.5, 8.2, 11.4, 11.0, 12.6. Put shortest unbiased 95% confidence limits on the parametric mean and variance of egg weight for this species.
25. Sputter coated specimens for scanning electron microscope examination deteriorate after usage. This deterioration manifests itself by isolated spots, within the field of view at any specific magnification, that charge. Any charging spot within a field of view ruins an electron micrograph made from that location. These spots occur independently of each other in space. Random examination of a specimen indicates that there exist on average 0.3 charging spots per field of view. A student, under time constraints decides to randomly choose a field of view and make an electron micrograph no-matter-what. What is the probability that this electron micrograph will not be spoiled by the presence of such a spot.
26. A manufacturer of plastic beads provides information concerning their mean diameter and standard deviation. Unfortunately, a biochemist has smudged the label containing these beads and can't read the mean, but can read the standard deviation to be 100 microns. The biochemist decides to estimate the parametric mean by measuring a sample of these beads. How large of a sample (i.e. how many beads) would he have to measure to estimate the parametric mean within 10 microns at 95% confidence.
27. A biology graduate student collects a sample size of n = 200. He is concerned that it might not be normally distributed so calculates g1 = 0.7 and g2 = 1.3. Is there a reason to reject the null hypothesis that these data are normally distributes? Use the 95% criterion.
28. Spiders disperse by ballooning on wind currents; this is effectively a random process. They impinge on a mangrove island off the coast of Florida at an average rate of 1 per hour. This has been established over a 3 year period during daylight hours. A young ecologist , Joe Buck, spent an hour sampling this process at night and during that time noted 4 spider immigrants. He concluded that dispersal rate in this system is higher at night. Do you agree? Provide the necessary statistical work to support your decision.
29. Dogma is that it is not important which codon of any set of redundant codons is used to code for an amino acid. Arginine has six codons (four beginning with a C and two with an A). Human IF-alpha-1 interferon uses C-type codons once for Arginine while it uses A-types ten times. Using your statistical knowledge comment on the dogma. Show work.
30. Endrin is a chemical used to protect directly sown Douglas Fir seed from seed-eating rodents. Although amounts of Endrin introduced into the forest environment are small, it is a very toxic pesticide to Coho salmon and other aquatic life. Concentrations high as 0.27 parts per billion for any extended period of time is a hazard to salmon. Soon after direct seeding of Douglas Fir, Endrin concentrations in ten random water samples were: .10, .08, .03, .15, .01, .09, .05, .17, .20 and .13. Estimate the mean concentration of Endrin with a 95% confidence interval.
31. A parasite has been described as being 300 microns long with a standard deviation of 50 microns. These values have been obtained from measurements solely obtained by collecting these larvae from the gut of laboratory white mice. A single individual was examined from a wild deer mouse and observed to be 180 microns long. Does this indicate host species significantly effects parasite morphology? Show work.
32. The mean maximum oral temperature reached during the course of a certain infectious disease was determined for a sample of 25 patients. The sample mean was 102° F with a sample standard deviation of 0.5 degree. Using 95% confidence limits, estimate the mean maximum temperature that you would expect in the population consisting of all patients who could have the disease.
33. Intraocular pressure is distributed normally in the population with a mean of 16 mm and a standard deviation of 3 mm. If a pressure of 21 mm or higher is considered abnormal, what percentage of the population have abnormally high intraocular pressure.
34. An investigator notices that children develop chronic bronchitis in the first year of life in 3 of 20 households where both parents are chronic bronchitis, as compared with the national incidence rate which is 5% in the first year of life. Is this difference real or can it be attributed to chance? Specifically, how likely are infants in at least 3 out of 20 households to develop chronic bronchitis if the probability is 0.05.
35. Professor Dill Dock provides his students with the sample data reproduced below and requests them to provide a 95% confidence interval for the parametric variance under the requirement that the parametric variance will exceed the upper limit with the same probability that it might be exceeded by the lower limit. Student Whiz Kid suggests that the range defined using the requirements imposed by Doctor Dock is larger than if the requirement was removed. Show what Whiz Kid had in mind by providing upper and lower confidence limits each way. How much larger was Dill Dock's range than Whiz Kid's.
17, 24, 18, 19, 30, 24, 25, 13, 19, 22
36. Female butterflies may repeatedly mate. Males deposit spermatophores on the female's sperm storage organs. By dissecting a female one can count the number of matings each female has engaged in. Researchers are interested in knowing if some females engage in a disproportionate number of matings or if those females mating frequently are simply the result of them randomly encountering more males. The following data exist from dissections.
# females 17 45 43 17 5 1 0 2
# spermatophores 0 1 2 3 4 5 6 7
What are the absolute expected frequencies for these outcomes?
37. Wheat height is in part the result of polygenic inheritance. Breeders determine the number of genes involved, in part, by assessing the variance of the F2 generation. A well studied strain has been determined to have a variance of 7.3 cm in height within the F2 generation. This strain has been treated with a mutagen and a sample of 25 F2 has a variance of 8.7 cm. Should the breeders strongly suspect that the number of genes involved in inheritance of this trait has changed.
38. The following results give plasma digoxin levels (in ng/ml) for 16 patients currently receiving a daily digoxin dose of 0.25 mg. Put shortest unbiased confidence limits on the mean and variance.
1.65 .60 .75 .95
.85 .55 .65 1.30
.70 .80 .90 .80
.95 1.10 .85 .60
39. Anax junius, a large dragonfly, has been studied frequently as a model predator. It has been suggested that it differentially eats larvae of the dragonfly Parithemis more often than equally sized larvae of Eurythemis. An experiment was conducted in which a very large number of Parithemis individuals and an equally very large number of Eurythemis individuals were made available as prey to Anax in a mixed culture. Data were collected on the first 10 individuals eaten by Anax, 8 were Parithemis and two were Eurythemis. Propose a null hypothesis (and alternative hypothesis) to test which will shed light on whether Anax is showing any preference for Parithemis. Can this hypothesis be rejected based on the data?
40. Auxotrophic microorganisms are mutant forms that can't synthesize certain molecules that wild type forms can. A certain laboratory has developed techniques by which an average of 2.1 colonies of these auxotrophs can be isolated every time the procedure is followed. These techniques have been in use for a considerable period of time. One researcher proposes an alternative mutagenic agent for producing greater numbers of colonies. He provides evidence of 4 auxotrophic colonies being produced in a test trial he ran. Evaluate his claim using statistical techniques you are familiar with.
41. The average shell length of bivalves in a local population is 45 mm with a standard deviation of 5. A sample of bivalves was obtained from the water column which indicated that they were dispersing; this sample included 25 individuals and had a mean shell length of 41 mm with a standard deviation of 4. Were these dispersing bivalves a representative random sample from the population? Show work.
42. Weight loss is a symptom of laboratory animals being parasitized. In one study the following data were obtained:
Individual # 1 2 3 4 5 6 7 8 9 10
Wt. loss (grams) 10 15 11 7 24 13 13 15 2 11
Develop 99% confidence limits for the mean and variance of weight loss. Make these shortest unbiased estimates.
43. Suppose the average length of stay in a chronic disease hospital for a certain type of patient is 60.0 days with a standard deviation of 15.0 days. Suppose many samples of 16 patients are selected at random with replacement from the hospital. For each sample the average length of hospital stay is computed. What two limits would include 99% of all sample means?
44. Wellborn and Robinson (in press) examined the leaf axils of 49 aquatic plants of Sagittaria platyphylla to see if larvae of the dragonfly species Pachydiplax longipennis were present in them. A total of 35 larvae were found in all these plants. Circumstantial evidence for interference for space between these larvae would exist if more plants had a single larva within their axils than would be expected by chance. How often would a single larva be expected to be found on one of these plants if their distribution among plants was random?
45. A medicine cabinet contains two similar bottles (A and B) each containing equal very large numbers of pills. Two tablets from either bottle A or B have no adverse effect. However, if a person ingests both a tablet from bottle A and one from bottle B within a 24 hour period a severe reaction is likely. Suppose a child dumps out both bottles and mixes the tablets up and consumes 2 of them. What is the probability of a severe reaction?
46. Suppose that the number of days until discharge for United States hospitals for patients with rheumatic fever is a positively skewed distribution, with a known to be 12. A random sample selected of 36 patients results in an average time until discharge of 22 days. Compute a 95% confidence interval for .
47. Baby Doc University Hospital applies for accreditation in the United States. Historical records from Baby Doc U.indicate patients with this condition have been released after a mean period of 15 days. Based on these data, are the standards of Baby Doc U. for releasing these patients less stringent than U.S. Hospitals?
48. Suppose that a group of 100 males ages 60-64 received a new flu vaccine in 1959 and that 5 of them died within the next year. Is this event unusual or can this kind of death-rate be expected for people of this age-sex group? From a 1959 U.S. life table, we find that 60-64 year old men have an approximate probability of death the next year of 0.028.
49. Height and weight are often used in epidemiological studies as possible predictors of disease outcomes. It is considerably less expensive and more convenient to use weights reported by the individuals themselves, but there is some concern that people will report lower weights than their actual weight. A study was performed to assess this with a sample of 10 individuals. Perform an analysis to determine if there is such a problem with self reported weights. The data follow:
Person # Self-reported weight Measured weight Difference
1 120 125 -5
2 120 118 +2
3 135 139 -4
4 118 120 -2
5 120 125 -5
6 190 198 -8
7 124 128 -4
9 133 131 +2
10 125 125 0
50. Is there evidence of digit preference among the self-reported weights in problem #2? Specifically, compare the observed proportion of reported weights whose last digit is 0 or 5 with the expected proportion based on chance and report how extreme the result is.
51. Put 95% confidence limits on the parametric mean and variance for the measured weights in problem #2.
52. Two experimental designs were proposed to evaluate the claim by a pharmaceutical company that their proposed drug decreases blood pressure by 10 units with a standard deviation of 7. Version #1 suggests evaluating the claim against an alternative of a 9 unit reduction in blood pressure by using a sample size of 500 subjects. Version #2 suggests evaluating the claim against an alternative of a 7 unit reduction in blood pressure by using a sample size of 1000. Calculate the power for each of the proposed tests.
53. Researchers were interested in evaluating if there might be a pattern between the concentration of plasma antioxidant vitamins and cancer risk. From a previous, very large study they knew that the concentration of plasma vitamin-A in healthy people had a mean of 2.88 micromoles/L and a standard deviation of 0.48. Their protocol involved evaluating the vitamin-A concentration in individuals with stomach cancer to determine if it was significantly less than that for healthy people. They soon began trying to determine the design strategy and could not decide whether their design should be able to detect a 5% (alternative mean = 2.73) or a 10% (alternative mean = 2.59) difference in concentrations. They decided that if they wanted to detect a 10% difference they would collect a sample size of 20, and therefore should use a sample size of 40 to be able to detect the smaller difference. They were willing to accept a type I error of 0.05, but did not know how to calculate the comparable power for their alternative approaches. Do this for them and decide which protocol has the greatest power.
54. A recent technique has been reported that permits humans to influence the gender of their offspring. Purportedly, this technique permits couples desiring male offspring a 70% chance of success. Couples are charged $5000/try for application of this technique which requires a moderate amount of laboratory manipulation and must pay regardless of success. The first 10 couples who used this procedure produced a total of 3 boys and 7 girls. Sensing blood, a lawyer has procured your services to demonstrate that this procedure does not fulfill its stated specifications. Perform an analysis to assess the procedure.
55. An unusually large number of undiagnosed stomach cramping complaints have been reported in New York City Public Schools during the past week. The distribution of case-frequencies among schools are presented at the end of this paragraph. Compare this distribution with one based simply on a random pattern to see if there is any pattern suggesting tat the disease might be contagious with some schools experiencing more and some less than expected. Do not go beyond establishing the theoretical expected frequencies (i.e., do not do goodness of fit test).
# schools 0 13 25 11 23 45 13 2
# of cases 1 2 3 4 5 6 7 >8
56. Sperm lengths in Telebasis salva, a coenagrionid damselfly, were measured and their dimensions are reported in a manuscript. The authors desired to put shortest unbiased 99% confidence intervals on the estimates of the parametric mean and variance. Please help them with these intervals. The data follow.
Length (in microns): 24, 26, 25, 26, 21, 21, 30, 26, 28
57. A physical therapist wishes to estimate with a 99 percent confidence the mean maximal strength of a particular muscle. Fifteen individuals are sampled and found to have a mean strength of 84.3 and a variance of 144. Provide unbiased 99% confidence limits on the parametric mean and variance.
58. One of the most common laboratory tests performed on any routine medical examination is a blood test. The two main aspects to a blood test are (1) counting the number of white blood cells (referred to as the "white count") and (2) differentiating the white blood cells that do exist into five categories, namely, neutrophils, lymphocytes, monocytes, eosinophils, and basophils (referred to as the "differential").
Evaluate probability of 2 lymphocytes out of 10 white blood cells if the probability that any one cell is a lymphocyte is 0.2.
If 10 cells were sampled and 7 were lymphocytes, would that cause you to reject the hypothesis that the probability was 0.2? (assume appropriate sampling techniques were used).
59. The mean serum creatine level measured in 12 patients 24 hours after they received a newly proposed antibiotic was 1.2 mg/dL. If the mean and standard deviation of serum creatine in the general population are 1.0 and 0.4 mg/dL, respectively, then using a significance level of 0.05, test if these 12 patients differed from the general population.
60. A public health issue arose concerning the possible carcinogenic potential of food ingredients containing ethylene dibromide (EDB). A study was performed looking at the mortality experienced by 161 individuals. Seven of these individuals died due to cancer during the study period. For this time period 5.8 cancer deaths were expected as calculated from the overall mortality rate in the U.S.A. Assess if the observed number of cancer deaths was excessive in this study group.
61. For a product to be advertised as being virucidal, the EPA runs a test in which they mix the proposed ingredient with polio virus and then add the suspension to animal cell cultures. A series of four 10% dilutions are made of the cell culture (i.e., 10% of the total solution, then 10% of that dilution, and 10% of that dilution, etc.). If any infective virus particles are observed from these dilutions the products claims are rejected. Assume the EPA can always find an infective particle if there is one in a dilution sample.
The worst case would be that the chemical inactivates all but 1 virus in the original suspension. How many analogous in-house tests showing zero infective particles would have to be run by the company for it to have a 95% confidence that its product is inactivating all viruses and will therefore pass the EPA test.
62. A radioactive source is emitting, on the average, one particles per minute. If counting continues for several hundred minutes, during which time the particles are emitted randomly, in what proportion of those minutes would you expect that something other than one particle would be emitted.
63. A cytologist has studied chromosome sizes in a large number of healthy persons, and found that for one particular chromosome the ratio of its long arm to its short arm is normally distributed with mean value 1.75 and variance of 0.0025. He measures the same ratio for the same chromosome in a patient suspected to have some genetic abnormalities; the value of the ratio is 1.61. Shall he classify the patient as healthy or not?
64. During the most recent period of glaciation the island Trinidad was connected with mainland South America. A well studied species of frog was known to have an adult average cranial volume of 0.36 cc with a variance of .005 on the mainland. A recent expedition on Trinidad collected 10 of these adult frogs which had a mean volume of 0.36 cc and a variance of .001. The researchers argued that the lower variance indicated stabilized selection on Trinidad. Does a statistical analysis provide any support for this argument?
65. As a preliminary experiment to evaluate the influence of buffer on the efficiency of a virucidal active ingredient, a test was performed using 5 different buffers and the same active ingredient. The logarithmic reduction in viral titer after a 10 minute exposure to the ingredient in each buffer was determined on two separate occasions for each buffer combination. The data follow:
Buffer A B C D E
Test 1 4.67 5.17 5.0 5.17 4.17
Test 2 5.0 5.0 5.27 5.0 4.5
Is there a significant ( = .05) effect of buffer? After obtaining these results the researchers wanted to know if buffer E was significantly inferior (i.e, different) than the other 4 buffers as a group. Please provide me with the answer.