Biometry Final Exam

 

1. Of 18 left handed children, 12 were female. But 24 of 52 right handed children were female. Are gender and handedness independent?

 

2. The wing lengths and tail lengths among birds of a particular species are:

                Wing length (cm)                           Tail length (cm)

10.4            7.4

10.8            7.6

11.1            7.9

10.2            7.2

10.3            7.4

10.2            7.1

10.7            7.4

10.5            7.2

10.8            7.8

11.2            7.7

10.6            7.8

11.3            8.3

 

Do not transform the data. Is there a significant correlation between these variables?  Is this correlation significantly different from 1.0.

 

3. Assume that an allometric analysis is performed on the above data. Do whatever transformation is appropriate and approximate the least square linear regression equation for wing on tail length.  Is this significant? What is the geometric mean regression coefficient for wing on tail? Does the last point qualify as an outlier?

 

4. The accompanying table gives the dry weight of 11 chick embryos ranging in age from 6 to 16 days. Calculate the least-squares estimates of the parameters of the regression line. Is the slope significantly different from zero. Find a 95% confidence interval for the true slope. Estimate the weight of embryos at age 8.5 days and put 95% confidence limits about this estimate.

 

Age                           6                7              8                9              10             11           12             13           14             15           16

Dry Weight                .029                .052                .079                .125                .181                .261                .425                .738                1.130                1.882                2.812

 (grams)

 

5. A study was done on the deaths of all men aged 50-54 in the country of Freedonia over a one month period. It was found that of the 55 people who died of cardiovascular disease, 15 were on a high salt diet before they died, whereas of the 45 people who died from other causes, 12 were on such a diet. Test for a relationship between the cause of death and type of diet.

 

6. Forced expiratory volume (FEV) is a standard measure of pulmonary function. To identify people with abnormal pulmonary function; standards of FEV for normal people must be established. It has been proposed that FEV is associated with height. Determine if the variables are associated and what percentage of the variation of FEV can be attributed to its association with height. Give 95% confidence limits for the parametric correlation coefficient.

 

Height (cm)                134                138                142                146                150                154                158                162                166                170    174    178

Mean FEV                1.7                1.9                2.0                2.1                2.2                2.5                2.7                3.0                3.1                3.4                3.8    3.9

(liters)

 

 

 

 

 

7. Communities of organisms were assembled over 6 month periods of time to assess whether habitats open to repeated invasion would reach an equilibrium pattern in which the average number of species showed neither a tendency to increase or decrease through time.  The last two months of data are presented below.  Do these data support the assertion that there are no such tendencies?

# species                7.0                7.5                6.9                6.8                6.5                -                6.5                6.6

week                       1                2                3                4                5                6                7                8

 

Notice that a datum is missing for week 6 due to a national meeting conflicting with science during that week.  Estimate the number of species that would be expected for that week and put confidence intervals about this estimate.

 

8. Researchers studying microbes in nearby lakes collected samples and made a variety of measurements on these samples.  These included estimating the relative frequency of dividing cells and the total chlorophyll content of the samples.  Are these two measurements significantly correlated?

Sample #                                               1                2                3                4                5                6

Frequency of dividing cells                .7                .3                .5                .7                .6                .3

Total chlorophyll                                 130                111                125                120                122                95

 

9.  The rates of oxygen consumption for tissue slices from nine species of mammals is provided below.  Determine if the correlation between body mass and relative rate of oxygen consumption by the brain statistically differs from the corresponding correlation between body mass and kidney.  Do not transform.

Oxygen Consumption

Species                                  Body Mass (kg)                             Brain                     Kidney

Mouse                                    0.021                                     32.9                       46.1

Rat                                         0.21                                       26.3                       38.2

Guinea Pig                           0.51                                       27.3                       31.8

Rabbit                                    1.05                                       28.2                       34.5

Cat                                         2.75                                       26.9                       22.7

Dog                                        15.9                                       21.2                       27.0

Sheep                                     49                                           19.7                       27.5

Cattle                                     420                                        17.2                       23.5

Horse                                     725                                        15.7                       21.5

What is the coefficient of determination for each case?

 

10.  A long term experiment storing 300 seeds from a species of plant and then at intervals of 5 years attempting to germinate 100 of these seeds to test for viability.  After 5 years, 75 of 100 germinated, after 10 years, 78 of 100 germinated, and after 15 years, 32 of the remaining 100 could germinate.  Within the time scale of the experiment, is germination independent of seed age?

 

11.  Annual primary productivity in North American hot deserts is a function of precipitation.  Determine the regression equations for the Mojave and Sonoran Deserts.  Are their slopes equal?  Are the relationships significant?  Is 1989 in the Mojave Desert too influential based on rules we discussed in class?

 

Mojave

Year                                       1985                      1986                      1987                      1988                      1989

Precipitation (inches)  3.2                          1.7                          4.5                          4.7                          15.1

Productivity (index)    117                        101                        155                        142                        201

 

                                                                Sonoran

Year                                       1985                      1986                      1987                      1988                      1989

Precipitation (inches)  17.1                       14.0                       23.0                       19.1                       24.0

Productivity (index)    250                        214                        318                        301                        325

 

 

 

 

12.  On any date adult female I. Posita display considerable between individual variations in mass.  There is even more variation between females collected early in the flight season and those collected later in the season.  Egg size seems to be associated with female size in an allometric fashion.  Make the appropriate transformation and determine the within date correlation of female size and egg size, as well as the between date correlation.  Are these correlations significantly different?  Are they significant?

 

Within date data

 

Female size (mg)                 23                           21                           25                           26                           20

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 Egg size (mg)                       15                           14                           16                           16                           13

 

 

Between date data

 

 Female size (mg)                33                           28                           22                           15                           14

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 Egg size (mg)                       21                           20                           14                           12                           12

 

 

13.  Notonecta hoffmani goes through five instars during its development.  Individuals from each instar were collected, marked and released at an aquatic habitat.  A recapture sample was obtained.  If we assume that this recapture procedure is random and the relative frequency of recapture reflects survivorship, is survivorship instar dependent?

 

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Instar                                     Marked                                 Recaptured

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1                                             150                                        102

2                                             75                                           57

3                                             55                                           50

4                                             40                                           35

5                                             45                                           40

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14.  A graduate student was interested in determining if the bat community structure in Big Bend National Park is different today than it was approximately 15 years ago.  She used mist nets and collected individuals during a summer field season and wanted to compare the number of individuals of each species she collected to the abundance distribution reported in a 5 year study that began in 1980.  She realized that there might be year to year patterns in the earlier study, so first wanted to determine if the earlier abundance patterns were dependent on the year of collection.  Help her evaluate this possibility.

 

Species                  1980                      1981                      1982                      1983                      1984                     

Species A                75                           100                        85                           80                           100 

Species B                50                           75                           57                           55                           80

Other                     60                           80                           65                           50                           90

 

15.  Dragonfly larvae swim using jet propulsion. A study was performed to establish the influence of their maximum cross sectional area to the drag force that larvae experience moving through the water.  Establish whether there is a significant causual relationship of cross sectional area on drag.  Then provide estimates of the appropriate parameters that define this relationship.

 

Cross sectional area        35                27                15                20                21

Drag                                      10                8                15                7                6

 

 

16.  The relative frequency that dragonfly larvae jet propel in response to a constant stimulus was established for 5 individuals of Anax junius larvae at 10°C and at that same temperature their swimming speed was measured.  An analogous study was performed at 5°C.  Establish if there is a significant correlation between the swimming speed and tendency to swim at each temperature, and then establish if the correlations are significantly different.  No transformation is necessary.

 

                                               

                                                5°C                                                        10°C

                                Tendency              Speed                     Tendency              Speed

                                45                           13                           75                           20

                                50                           17                           65                           18

                                30                           8                             80                           25

                                85                           21                           80                           28

                                70                           20                           82                           30

 

17.  The following data were collected in a study of the relationship between intelligence and family size:

 

                                Number of children                                            Average measured intelligence score of all

                                                                                                                                children in the family

                                               1                                                                                             105

                                                2                                                                                             102

                                                3                                                                                             104

                                                4                                                                                             100

                                                5                                                                                             97

                                                6                                                                                             101

                                                7                                                                                             95

                                                8                                                                                             93

                                                9                                                                                             97

                                                10                                                                                           88

 

Develop the least squares regression equation to represent this situation.  Is the regression equation significant? (show work)?  What is the coefficient of determination?  What are the leverage coefficient and standardized residual for the 10 child family data point?  Does the point qualify as an outlier?  Put a 95% confidence limit around your estimate of the intelligence score of 4 child families.

 

18.  People may be characterized as either chronic carriers or non-carriers of a certain antigen.  A recent study of 650 individuals of 4 different ABO blood groups found 72 of 302 type O individuals were carriers, 54 of 246 Type A individuals were carriers, 16 of 79 Type B individuals were carriers and 8 of 23 Type AB individuals were carriers.  The research team felt that this demonstrated that the probability of being a carrier of this antigen was influenced by an individuals ABO blood group.  Do you agree (show work).

 

19.  The damselfly species Ischnura ramburi has dichromatic females, one type referred to as the andromorph, and the other as the heteromorph.  To determine if either of these morphs is more commonly engaged in reproductive behavior an ecologist collected seventy five females (75), 53 of which were in copula.  Of these 53, 38 were heteromorphs.  Of the 22 females not  in copula, 10 were heteromorphs.  Is there any evidence from these data that mating frequency of I. ramburi females is dependent on morph type.

 

 

 

 

 

 

 

 

20.  Assume that different species can be considered independent data points in the following problem.  Determine if size-sexual dimorphism in ischnuran damselflies is size dependent.  To do this, develop the allometric relationship using least square regression of male hind wing length on female hind wing length for the following 5 species.  Is the slope significantly different from 1?  What would be the slope of the Geometric Mean Regression?

 

 Male                      11                           14                           13.5                       14                           15.5

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Female                   14.5                       17.25                     15.25                     15                           16

 

 

21.  Develop the correlation coefficient for the logarithmically transformed values for these other 7 ischnuran species.  Is it different from the correlation coefficient developed in problem 2?

 

Male                       10.5                       12                           12.5                       15                           16               17.5                        16

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Female                   13.25                     14.5                       14.25                     17                           17.25        19.25                17.25

 

 

22.  The rates of swimming trials were made between dragonfly larvae belonging to 4 species.  These individuals were initially screened to ensure they were of equal weight therefore size differences did not effect the outcome.  From the results of this experiment, I would like to be able to compare any 2 species and comment on whether their comparative speeds are significantly different.  Data are in cm/sec and were obtained at 20°C.

 

A. junius                P. tenera     P. Lydia                 L. luctosa

                14                           7.5                          8.9                          4.5

12.7                        8                             7.5                          5.1

12.9                        9.7                          8.8                          3.7

14.2                        10.1                       8.5                          5.2

13.4                        7.7                          8.0                          5.1

 

23.  To determine if swimming speed is temperature dependent the same 5 A. junius were run in the same order at 25°C.  The following data were obtained: 15.7, 14.1, 14.9, 17.0, 13.1.  Is swimming speed temperature dependent in this species?

 

24.   Assume the normality assumption was violated in problem 2, determine if there is a significant between group effect using a nonparametric procedure.

 

25.  The following are the numbers of a particular species of organism found in 100 samples of water from a  pond.   Test the null hypothesis that these data were drawn from a Poisson distribution.

 

                                                                Number of Organisms per sample                                                                                   Frequency

                                                                                                0                                                                                                                                                             15

                                                                                                1                                                                                                                                                             30

                                                                                                2                                                                                                                                                             25

                                                                                                3                                                                                                                                                             20

                                                                                                4                                                                                                                                                             5

                                                                                                5                                                                                                                                                             4

                                                                                                6                                                                                                                                                             1

                                                                                                7                                                                                                                                                             0

 

 

 

 

 

 

26.  The following table shows the weight and total cholesterol and triglyceride levels in patients with primary type II hyperlipoproteinemia just prior to initiation of the treatment.  Test the null hypothesis that the correlation of weight and total cholesterol is equivalent to the correlation of weight and triglyceride levels.

 

Patient #                                               1                                2                                3                                4                                5                                6                                7

                ¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾

Weight (kg)                                         76                                97                                83                                52                                70                                67                                75

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Cholesterol (mg/100 ml)                                302                                336                                220                                300                                382                                379                                331

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Triglyceride (mg/100 ml)                                139                                101                                57                                56                                113                                42                                84

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27.  Marden (1987) collected flight data on the following 15 damselfly individuals:

 

                                                                                                                                                                                                Flight-muscle                                      Maximum Lift

                                Species                                                  Body Mass (g)                                          Mass (g)                                                Force (Nx102)      

¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾¾

Megaloptera sp.                                  0.233                                                                                     0.076                                                                                     0.57

Mecistogaster sp.                                                                0.149                                                                                     0.045                                                                                     0.32

                                Mecistogaster sp.                                                                0.217                                                                                     0.063                                                                                     0.51

Argia chelata                                      0.037                                                                                     0.017                                                                                     0.15

Argia chelata                                      0.035                                                                                     0.016                                                                                     0.11

Argia chelata                                      0.024                                                                                     0.011                                                                                     0.09

Argia chelata                                      0.022                                                                                     0.010                                                                                     0.09

Argia chelata                                      0.030                                                                                     0.013                                                                                     0.15

Argia chelata                                      0.039                                                                                     0.018                                                                                     0.17

Argia chelata                                      0.046                                                                                     0.015                                                                                     0.13

Argia chelata                                      0.035                                                                                     0.016                                                                                     0.13

Argia chelata                                      0.019                                                                                     0.008                                                                                     0.07

Argia chelata                                      0.042                                                                                     0.018                                                                                     0.17

Argia chelata                                      0.055                                                                                     0.020                                                                                     0.15

Argia chelata                                      0.036                                                                                     0.015                                                                                     0.17

 

Ellington (1991) used these data to develop a regression(both dependent and independent variables log transformed) of muscle mass specific power (i.e. Max Lift Force/Flight Muscle Mass) on body mass.  Repeat Ellington’s analysis and provide estimates of a, b, r2, and significance of regression.  Since all data points but 3 are from conspecifics, determine the leverage and standardized residuals of these three points.  Are they influential in a statistical sense?

 

28.  During the past twelve years, a major university has produced graduate students having masters degrees according to the following table:

 

year                                1                                2                                3                                4                                5                                6                                7                                8                                9                                10                                11                                12

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#                                1                                2                                8                                4                                5                                7                                10                                4                                6                                7                                9                                7

 

                                                                Test for a significant regression between numbers and years using both raw data and transformed data in which the relationship 10y = axb is posited.  Evaluate constants and coefficients of determination for each.  If this was a correlation problem would r be significant?

 

 

 

 

29.  As part of a study of the microvascular complications of diabetes mellitus and their possible remediation through blood glucose control, the capillary basement-membrane width (in angstroms) and glycosylated hemoglobin level (as a pervcentage) of 13 diabetic patients were measured.  The values are reported here and require no transformation.  Test whether these results offer any evidence of a connection (provide the associated statistics and significance tests) between these two patient characteristics and measure the strength of the connection, if any.

 

                                Capillary Basement                                              Glycosylated Hemoglobin

Membrane width                                                                                  Level

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2642                                                                                                                      7.7

1180                                                                                                                      16.7

1474                                                                                                                      10.4

1798                                                                                                                      10.9

2958                                                                                                                      7.2

3480                                                                                                                      8.5

1917                                                                                                                      10.2

2359                                                                                                                      12.7

1723                                                                                                                      11.4

2414                                                                                                                      8.1

900                                                                                                                         11.6

2135                                                                                                                      8.4

2347                                                                                                                      8.3

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30.  A researcher in New York has used a sample size of 10 and reported a correlation coefficient of -0.55.  Do the two studies reach statistically different conclusions?

 

31.  Higginbotham (unpublished) sampled bats at Menagerie Springs in Big Bend National Park during three separate years (1996, 1997, 1998).  The numbers she collected during these three years respectively are included in parentheses following each of 5 species of bats: Antrozus pallidus (6, 20, 1), Tadarida brasilensis (103, 564, 5), Nyctinomops femorosaccus (39, 182, 16), N. macrotis (13,  35, 7), and Eumops perotis (23, 53, 5).  Was the relative sampling success for obtaining different species dependent on the year.