Paleontology: Diversity and Extinction, Taphonomy, Functional Morphology, Morphometrics, Vertebrates and Paleoecology
2. Mass extinctions: causes and effects
A mass extinction is a geologically brief interval (usually under a few Myr) in which a large share of species worldwide disappears, far above the background rate, across unrelated groups and habitats.
| Event | Age (Ma, approx.) | Marine species lost (approx.; estimates vary) | Main proposed causes |
|---|---|---|---|
| End-Ordovician | 445 | about 85% | Gondwanan glaciation with sea-level fall, then warming and anoxia |
| Late Devonian | 372 | about 75% | Repeated anoxic pulses, climate change, collapse of reef ecosystems |
| End-Permian | 252 | about 90% or more | Siberian Traps volcanism, warming, ocean anoxia and acidification |
| End-Triassic | 201 | about 75 to 80% | Central Atlantic Magmatic Province volcanism, CO₂ warming, acidification |
| End-Cretaceous | 66 | about 75% | Chicxulub impact, with Deccan Traps volcanism (relative roles debated) |
At the end of the Cretaceous the evidence for impact is an iridium-enriched clay layer, shocked quartz, spherules and the Chicxulub crater (Yucatán, about 180 km across). Volcanism acts through CO₂ (long-term warming and acidification), SO₂ aerosols (short-term cooling) and warm, stagnant, oxygen-poor oceans. The end-Permian removed the trilobites and rugose and tabulate corals, so the Paleozoic fauna never regained dominance.
Effects. Extinction is selective: geographically restricted, narrowly adapted, tropical reef-building and large-bodied taxa suffer most. Survivors radiate into the emptied ecospace (mammals after the end-Cretaceous, archosaurs after the end-Permian), but marine recovery takes several million years, and disaster taxa (the bivalve Claraia and the therapsid Lystrosaurus after the end-Permian) briefly dominate. A Lazarus taxon vanishes from the record and reappears later. Because a genus survives if any one of its species does, the percentage of species lost normally exceeds the percentage of genera lost in the same event.
3. Taphonomy: processes of fossilisation
Taphonomy studies everything that happens to remains between death and discovery. Biostratinomy covers death to final burial (scavenging, decay, disarticulation, transport, abrasion, bioerosion); fossil diagenesis covers what follows burial (compaction, cementation, dissolution, recrystallisation, replacement). Preservation needs durable hard parts, rapid burial and often oxygen-poor bottom water that keeps out scavengers.
- Permineralisation: pores fill with silica, calcite or pyrite while the original tissue stays (petrified wood, bone).
- Replacement: the original substance dissolves and another mineral takes its place molecule by molecule (silicified or pyritised shells).
- Recrystallisation or inversion: the composition is kept but the crystal form changes (aragonite to calcite), destroying fine structure.
- Carbonisation (distillation): volatiles escape and a thin carbon film remains (graptolites, leaves, soft tissue).
- Moulds and casts: a dissolved shell leaves an external mould (impression of the outside) and an internal mould or steinkern (the infilled interior); a void filled by cement gives a cast.
- Unaltered remains: amber (insects), freezing (mammoths), desiccation, tar seeps.
Mineralogy decides what survives. Aragonite (many bivalves and gastropods, scleractinian corals, ammonite shells) dissolves or inverts easily, so these groups often survive only as moulds, whereas low-Mg calcite (articulate brachiopods, trilobites), high-Mg calcite (echinoderm plates), apatite (bone, conodonts, Lingula) and silica (radiolarians, sponge spicules) are more durable. An assemblage may be autochthonous (in life position), parautochthonous or allochthonous (transported). Time-averaging mixes generations spanning hundreds to thousands of years into one bed, so a death assemblage is a long-term average, not a snapshot.
Lagerstätten are deposits of exceptional fossil content. Konzentrat deposits concentrate hard parts (bone beds, shell beds); Konservat deposits preserve soft tissue: the Burgess Shale (Cambrian, British Columbia), Chengjiang (early Cambrian, China) and the Solnhofen Limestone (Late Jurassic, Germany). The record is therefore biased towards hard-shelled, shallow-marine, rapidly buried organisms, and diversity counts inherit this bias.
4. Functional morphology of invertebrates
Functional morphology reads a skeleton as an engineering solution. The method is to compare with living relatives, model the structure, and test the model against the sediment and associated fossils.
| Group | Skeleton and symmetry | Functional features and what they tell |
|---|---|---|
| Bivalves | Two valves, mirror images of each other across the commissure plane; one valve alone is usually not symmetrical (inequilateral) | A deep pallial sinus (notch in the pallial line where siphon muscles attach) means long siphons and a deep burrower; a byssal notch means anchoring by threads; one valve cemented down means an oyster-type cementer |
| Brachiopods | Two unequal valves (pedicle and brachial); the symmetry plane cuts through both valves, and each valve is itself bilaterally symmetrical. Articulate forms have low-Mg calcite shells with teeth and sockets; inarticulate forms (Lingula) have phosphatic shells held by muscles only | The pedicle emerges through an opening in the pedicle valve to anchor the animal; internal brachidia (spiral in spiriferids) support the feeding lophophore; a wide hinge line stabilises the shell on soft mud |
| Gastropods | One univalve shell, usually coiled: trochospiral (helical) or planispiral. Torsion, a 180° twist of the body in the larva that brings the mantle cavity forward, is not the same as shell coiling | Coiling is dextral (aperture on the right when the apex is up and the aperture faces the viewer) or sinistral; a siphonal canal at the aperture is typical of carnivorous snails, while a rounded entire aperture is common in grazers |
| Echinoids | Calcite test of plates arranged in five ambulacral (tube-foot pores) and five interambulacral areas; radial in regular urchins, secondarily bilateral in irregular ones | Regular urchins live on the surface, with the mouth below, the anus (periproct) at the apex and a chewing Aristotle’s lantern. Irregular urchins (heart urchins, sand dollars) burrow: periproct shifted to the posterior, short dense spines, flat or heart-shaped test; heart urchins have lost the lantern |
| Ammonites | External chambered shell, usually planispiral; septa divide the chambers and a siphuncle runs along the outer (ventral) margin in ammonoids | The suture, where a septum meets the shell wall, is typically goniatitic (simple zigzag) in the Devonian–Permian, ceratitic in the Triassic and ammonitic (finely frilled) in the Jurassic–Cretaceous. Compressed, streamlined, involute shells swim efficiently, open evolute shells less so. Rapid evolution makes them Mesozoic zone fossils |
5. Microfossils: foraminifera, ostracoda, conodonts and bryozoa
Microfossils, mostly under a millimetre long, are recovered by sieving and acid residues; a small drill sample holds hundreds, so they are the working tools of subsurface biostratigraphy.
| Group | Test and range | Main uses |
|---|---|---|
| Foraminifera | Single-celled protists with a chambered test: agglutinated, porcellaneous or hyaline calcareous. Benthic forms from the Cambrian, planktic forms from the Jurassic | Planktic species zone the Jurassic to Recent; benthic assemblages and the planktic : benthic ratio (higher offshore) show depth; Neogloboquadrina pachyderma coils mostly sinistral in cold water and dextral in warm; δ¹⁸O of calcite tests gives temperature and ice volume |
| Ostracoda | Bivalved crustaceans with a low-Mg calcite carapace; valves unequal and hinged, with muscle scars; Ordovician to Recent | Ornament, thickness and assemblage composition indicate salinity, temperature and depth; Mg/Ca and δ¹⁸O of the shell give water temperature |
| Conodonts | Tooth-like phosphatic (apatite) elements of an extinct eel-like chordate, forming apparatuses of several element types; late Cambrian to end-Triassic | Fast evolving, so the main zonation of Ordovician to Triassic marine rocks. The colour alteration index (CAI 1, pale, to CAI 5, black) records irreversibly the maximum heating, from about 50 to 80 °C (CAI 1) to above 300 °C (CAI 5) |
| Bryozoa | Colonial animals; each zooid lives in a calcareous box or tube (zooecium) and feeds with a lophophore; Ordovician to Recent; mostly marine | Colony form tends to track water energy: encrusting sheets in turbulent water, erect branching or fan-like (fenestrate) colonies in calmer water; cheilostomes dominate Cenozoic shelves |
6. Statistical analysis of morphology
Allometry describes growth of one part relative to another by Huxley’s power law y = a x^b, which is a straight line of slope b on a log–log plot: log y = log a + b log x. Isometry (shape unchanged with size) means b equals the value geometry demands: 1 for two lengths, 2 for an area against a length, 3 for a volume or mass against a length. A larger b is positive allometry, a smaller b negative. The slope is ontogenetic within one growth series, static among individuals of one age and evolutionary among species. Example: a shell with x = 10 and y = 4 and another with x = 40 and y = 32 give b = ln 8 / ln 4 = 1.5, so at x = 90 the predicted y is 4 × 9^1.5 = 108.
For n measurements the mean is x̄, the standard deviation s, the standard error SE = s / √n and the coefficient of variation CV = 100 s / x̄. A 95% confidence interval for the mean is x̄ ± t s / √n, with t from Student’s distribution at n − 1 degrees of freedom: for n = 16, x̄ = 42.0 mm, s = 6.0 mm and t = 2.131, the half-width is 2.131 × 6 / 4 = 3.20 mm. Quadrupling the sample only halves the interval. A bimodal size histogram suggests two cohorts, sexual dimorphism or two species, which must be tested, not assumed.
Shape is compared with multivariate methods: principal components (the first axis usually carries size), discriminant analysis, and landmark-based geometric morphometrics, which removes position, scale and rotation before shape is compared. Raup’s theoretical morphology describes a coiled shell by W (whorl expansion rate), D (distance of the generating curve from the axis) and T (translation along the axis, zero in planispiral shells).
7. Vertebrate paleontology
The vertebrate story is a series of transitions, each recognised from a few key anatomical changes.
| Transition | Key evidence |
|---|---|
| Fish to tetrapods (Late Devonian, about 375 to 360 Ma) | Lobe-finned fishes with limb-like fin bones; Tiktaalik (neck, wrist-like fin) then Acanthostega and Ichthyostega with digits |
| Amniotes | The amniotic egg frees reproduction from water (Carboniferous). Skull openings classify them: anapsid none, synapsid one lower temporal opening (therapsids, mammals), diapsid two (lizards, snakes, crocodiles, dinosaurs, birds) |
| Therapsids to mammals (Triassic) | Cynodont ancestors; a single lower jaw bone (dentary) joined to the squamosal; the old jaw-joint bones (articular, quadrate) become the middle-ear malleus and incus; teeth differentiate into incisors, canines, premolars and molars |
| Dinosaurs and birds | Saurischians are lizard-hipped (theropods, sauropodomorphs); ornithischians are bird-hipped (pubis swung back beside the ischium). Birds descend from theropod saurischians; Archaeopteryx (Late Jurassic, Solnhofen) has feathers, teeth and a long bony tail. Pterosaurs, ichthyosaurs and plesiosaurs are not dinosaurs |
| Whales | Eocene Pakicetus and related forms from the Indo-Pakistan region show the land-to-sea transition |
Teeth and limbs read diet and movement: sharp shearing teeth in carnivores; flat, high-crowned (hypsodont) cheek teeth in grazers, which spread with Miocene grasslands. India has a rich record: the Late Triassic Maleri Formation (rhynchosaurs, phytosaurs), the Early Jurassic Kota Formation (the sauropod Barapasaurus), the Late Cretaceous Lameta Formation (titanosaurs, Rajasaurus, eggs) and the Siwalik Group (Miocene to Pleistocene mammals, the ape Sivapithecus).
8. Paleoecology, fossils and paleoenvironments
Paleoecology reconstructs how ancient organisms related to their surroundings. Its basic terms are life habit (benthic epifauna or infauna, nekton, plankton), feeding mode (suspension, deposit, grazing, carnivore), community (organisms that lived together) against assemblage (what was buried together, which may mix communities through transport and time-averaging). The method is uniformitarian: use nearest living relatives, functional morphology and the enclosing sediment, and prefer several independent indicators.
| Evidence | What it indicates |
|---|---|
| Hermatypic (zooxanthellate) reef corals | Warm, clear, shallow water of normal salinity within the photic zone |
| Stenohaline groups (echinoderms, brachiopods, cephalopods, corals) | Normal marine salinity; their absence, with a few abundant euryhaline species such as some bivalves, gastropods and ostracods, points to brackish or hypersaline water |
| Laminated black shale, no bioturbation, no benthic shells | Oxygen-depleted bottom water |
| Mostly entire-margined leaves in a woody dicot flora | Warm, typically tropical climate; more toothed leaves mean cooler |
| Trace fossils: Skolithos (vertical dwelling burrows), Cruziana (horizontal feeding and resting traces), Zoophycos, Nereites (patterned grazing traces) | Ichnofacies: Skolithos, clean mobile sands of a high-energy shoreface; Cruziana, quieter lower shoreface to shelf; Zoophycos, quiet deeper muddy water; Nereites, deep-sea turbidite floors |
Index fossils date rocks. They must be widespread, abundant, fast-evolving (short range), easy to identify and not restricted to one facies. Facies fossils are the reverse: long-ranging and tied to a habitat, so they show environment rather than age. A good paleoenvironmental indicator is therefore usually a poor index fossil, and the reverse.
Key takeaways
- Diversity moves through three evolutionary faunas (Cambrian, Paleozoic, Modern). Compare richness only after rarefaction; Shannon H = −Σ p ln p peaks at ln S, Simpson D = Σ p²; an extinction rate is per capita, q = −ln(N_bt/N_b)/Δt, not a bare percentage.
- The Big Five (about 445, 372, 252, 201 and 66 Ma) are the end-Ordovician, Late Devonian, end-Permian (largest), end-Triassic and end-Cretaceous. Extinction is selective, species loss normally exceeds genus loss, and the Signor–Lipps effect makes sudden events look gradual.
- Taphonomy runs from biostratinomy (death to burial) to diagenesis. Aragonite shells dissolve and leave moulds, time-averaging blurs assemblages, and Konservat Lagerstätten (Burgess Shale, Chengjiang, Solnhofen) preserve soft tissue while Konzentrat deposits only concentrate hard parts.
- Bivalve valves are mirror images across the commissure and each is normally asymmetrical; brachiopod valves are each symmetrical and unequal. A deep pallial sinus means a deep burrower, torsion is not coiling, and ammonoid sutures run goniatitic, ceratitic, ammonitic.
- Allometry is y = a x^b, slope b on log–log axes, compared with the isometric 1, 2 or 3; r is not the slope. A mean’s 95% interval is x̄ ± t s/√n, so quadrupling n halves it.
- Vertebrate history runs fish, tetrapod, amniote, therapsid, mammal, with birds nested in theropod (saurischian) dinosaurs, not the bird-hipped ornithischians. Index fossils are widespread, short-ranging and facies-free; facies fossils and ichnofacies (Skolithos, Cruziana, Zoophycos, Nereites) read environment.
Practice questions (18)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
A clay layer at a stratigraphic boundary contains an iridium anomaly, shocked quartz grains and spherules, and a buried crater about 180 km across has been dated to the same time. Which mass extinction is this evidence most directly associated with?
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Answer: D — The end-Cretaceous extinction, attributed mainly to a large impact at Chicxulub
Iridium is rare in crustal rock but abundant in meteorites, and shocked quartz, spherules and a crater of about 180 km are the signature of a large impact; this set of evidence belongs to the end-Cretaceous boundary at about 66 Ma. The end-Permian is the tempting wrong answer because it is the largest extinction, but its record points to flood-basalt volcanism and ocean chemistry, not shock features. The end-Triassic and end-Ordovician events are likewise tied to volcanism and glaciation.In a measured section the last occurrences of 12 ammonite species are spread through the 4 m of rock below a boundary clay, although independent evidence shows that all 12 were eliminated in one brief event at the boundary. Which effect best explains the apparent gradual disappearance?
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Answer: A — The Signor–Lipps effect: with incomplete sampling the last preserved fossil of a taxon lies below its true last occurrence, so ranges seem to end early
Incomplete preservation means the last fossil found lies below the true last occurrence, so a single event is smeared into an apparent gradual decline: this is the Signor–Lipps effect. A Lazarus taxon needs a reappearance above the gap, which is not described here. The pull of the Recent concerns the better sampling of young rocks and exaggerates diversity near the present. Time-averaging would push apparent last occurrences upward (older shells in younger beds), the opposite of the smearing below the boundary.A mass extinction removes 60% of the genera of a marine group. Species were lost at random, with the same probability in every genus, and most genera of the group contain several species. Which statement about species loss is correct?
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Answer: C — Species loss was well above 60%, because a genus disappears only when all of its species have disappeared
A genus is lost only if all k of its species are lost. With a random species loss fraction p the genus loss fraction is p^k, which for k > 1 is smaller than p. A genus loss of 60% therefore needs p^k = 0.6, so p = 0.6^(1/k) is well above 0.6 (for k = 3, p is about 0.84). The tempting answer, species loss below 60%, reverses this logic: surviving genera are the ones that retained at least one species, but extinct genera lost all of theirs, which makes the species percentage the larger.A sandstone bed contains many cavities shaped like gastropod and bivalve shells but no shell material, while the associated articulate brachiopod shells (low-Mg calcite) are intact. Which explanation is most likely?
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Answer: B — The aragonitic molluscan shells dissolved after burial and left moulds, while the low-Mg calcite brachiopod shells resisted dissolution
Aragonite is the less stable CaCO3 polymorph and is dissolved or inverted early in diagenesis, whereas low-Mg calcite is stable. Mollusc shells (aragonite) therefore survive as external and internal moulds beside intact calcitic brachiopods. Carbonisation leaves carbon films, not shaped cavities. Permineralisation preserves the shell material itself, so the cavities would not be empty. Time-averaging blends ages but does not selectively delete one mineralogy.A fossil shell has two unequal valves. A plane of bilateral symmetry cuts through the middle of each valve, and the larger valve has a small opening near its beak for a fleshy stalk. Which identification is best supported?
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Answer: D — A brachiopod, because each valve is symmetrical on its own and the pedicle emerges through the larger valve
In brachiopods the plane of symmetry passes through both valves, so each valve is symmetrical, the valves differ in size, and the pedicle leaves through an opening in the pedicle valve. Bivalve molluscs have a symmetry plane between the two valves, so a valve alone is normally asymmetrical. Oysters are inequivalve, which makes them the tempting answer, but their valves are irregular and they have no pedicle opening. A gastropod has one shell, not two valves.In a deep-sea core from the North Atlantic, the share of sinistrally coiled Neogloboquadrina pachyderma falls steadily from 90% at the base to 10% at the top, while the planktic : benthic foraminiferal ratio stays constant. Which interpretation is best?
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Answer: B — Surface water warmed upwards, because this species coils mainly sinistrally in cold water and dextrally in warm water
In Neogloboquadrina pachyderma the dominant coiling direction tracks surface temperature: sinistral in cold water, dextral in warm. A fall from 90% to 10% sinistral therefore records warming upcore. A depth change would normally shift the planktic : benthic ratio, which stays constant here, and coiling in this species is not a salinity or depth indicator. The option claiming that sinistral coiling marks warm water reverses the temperature relationship.Dinosaurs are divided into lizard-hipped and bird-hipped groups. Which statement about hip structure and the ancestry of birds is correct?
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Answer: A — Birds descend from theropod dinosaurs, which are lizard-hipped saurischians; the bird-hipped dinosaurs are the ornithischians
Birds evolved from small theropods, a lizard-hipped saurischian lineage; Archaeopteryx is a theropod-like bird of the Late Jurassic. The ornithischians are called bird-hipped because their pubis rotated backwards beside the ischium, a condition birds evolved independently, so the name misleads. Pterosaurs are not dinosaurs and became extinct at the end-Cretaceous.A mudstone unit contains millions of individuals of only three species, all thin-shelled bivalves, gastropods and ostracods. Echinoderms, brachiopods, corals and cephalopods are absent from this unit although they occur in the beds above and below it. Which environment is most likely?
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Answer: C — A brackish lagoon or estuary, where only euryhaline taxa tolerate the fluctuating salinity and can reach high abundance
Echinoderms, brachiopods, corals and cephalopods are stenohaline and need normal marine salinity. Their absence between beds that contain them, with a low-diversity, very abundant assemblage of tolerant bivalves, gastropods and ostracods, indicates fluctuating, brackish conditions such as a lagoon or estuary. A reef flat or an open shelf would carry the stenohaline groups, and the bed lies between marine beds, not a cold basin floor, so temperature alone does not explain it.Which description fits a Konservat-Lagerstätte?
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Answer: D — A deposit, such as the Burgess Shale or the Solnhofen Limestone, that preserves soft tissues or whole bodies in exceptional detail
A Konservat-Lagerstätte is defined by exceptional preservation of soft tissue or whole bodies, as in the Burgess Shale, Chengjiang and Solnhofen. A concentration of hard parts, such as a bone bed, is a Konzentrat-Lagerstätte, which is the tempting confusion. Moulds and mixed transported assemblages describe other taphonomic outcomes and carry no claim of exceptional soft-tissue preservation.Which of the following statements about mass extinctions are correct?
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Answer: A — Extinction is often selective: narrowly adapted, geographically restricted and reef-building taxa tend to suffer more than widespread generalists; C — Disaster taxa, such as the bivalve Claraia and the therapsid Lystrosaurus after the end-Permian event, briefly become abundant in the aftermath
Extinction is selective, and disaster taxa bloom after the end-Permian crisis, so those two statements are correct. A Lazarus taxon is one that disappears from the fossil record and later reappears, not one that is long-lived. Because a genus is lost only when all its species are, the percentage of genera lost is normally smaller than the percentage of species lost, so the genus-versus-species statement is reversed.Which of the following statements about invertebrate functional morphology are correct?
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Answer: A — Gastropod torsion is a 180° twist of the body mass during larval development and is distinct from the coiling of the shell; C — A deep pallial sinus in a bivalve indicates long siphons and a deep-burrowing habit; D — Ammonoid sutures are typically goniatitic in the Devonian to Permian, ceratitic in the Triassic and ammonitic in the Jurassic and Cretaceous
Torsion is a larval twist unrelated to shell coiling; the pallial sinus marks the attachment of siphon retractors, so a deep sinus means long siphons and deep burial; and the goniatitic, ceratitic, ammonitic sequence is the usual pattern of increasing suture complexity. Regular echinoids are radially symmetrical with the anus at the apex; it is the irregular echinoids that have a displaced anus and secondary bilateral symmetry, so that statement is wrong.Which of the following statements about vertebrate evolution are correct?
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Answer: A — Tiktaalik, a Late Devonian lobe-finned fish with a neck and wrist-like fin bones, is a transitional form between fishes and tetrapods; B — Synapsids, the group containing the therapsids and mammals, have a single opening in the skull behind the eye on each side; D — In the evolution of the mammalian ear, the articular and quadrate bones of the reptilian jaw joint became the malleus and incus
Tiktaalik is the classic fish-tetrapod transitional form; synapsids are defined by one lower temporal opening; and the reptilian articular and quadrate became the malleus and incus as the mammalian jaw joint shifted to the dentary and squamosal. Pterosaurs (flying archosaurs) and ichthyosaurs (marine reptiles) are not dinosaurs, so that statement is wrong.Which of the following statements about fossils and paleoenvironments are correct?
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Answer: B — A good index fossil is widespread, abundant, quickly evolving and not restricted to a single facies; D — Vertical dwelling burrows of the Skolithos type in clean sand typify a high-energy shoreface environment
An index fossil must be widespread, abundant, fast-evolving and independent of facies, and the Skolithos ichnofacies of vertical dwelling burrows marks high-energy sandy shorefaces. A death assemblage is usually time-averaged and may mix communities, so it is not a snapshot. Entire-margined leaves, not toothed ones, dominate warm climates, so the leaf statement is reversed.A bed contains four species with 40, 30, 20 and 10 individuals. Using Shannon’s H = −Σ p_i ln p_i, with p_i the proportion of individuals of species i and the natural logarithm, the value of H is ____ (give the answer to two decimal places).
Numerical answer — type the value.
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Answer: 1.28
p = 0.40, 0.30, 0.20, 0.10. The terms −p ln p are 0.4 × 0.9163 = 0.3665, 0.3 × 1.2040 = 0.3612, 0.2 × 1.6094 = 0.3219 and 0.1 × 2.3026 = 0.2303. Their sum is 1.2799, which is 1.28. The maximum for four species would be ln 4 = 1.39, so the bed is fairly even.In a stage lasting 4 Myr, 120 genera cross its bottom boundary. Of these, 30 also cross its top boundary and the other 90 become extinct within the stage. A further 40 genera originate and become extinct entirely inside the stage. Use q = −ln(N_bt / N_b) / Δt, where N_b is the number of taxa crossing the bottom boundary, N_bt the number crossing both boundaries and Δt the length in Myr. The per-capita extinction rate q, in per Myr, is ____ (give the answer to three decimal places).
Numerical answer — type the value.
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Answer: 0.347
N_b = 120 and N_bt = 30, so N_bt / N_b = 0.25 and −ln 0.25 = 1.3863. Dividing by Δt = 4 Myr gives q = 0.3466, which is 0.347 per Myr. The 40 genera confined to the stage cross neither boundary and are not counted, and using 90 (the number that went extinct) instead of the survivors is the usual slip.A sample of N = 10 fossil individuals contains three species with 6, 3 and 1 individuals. Two individuals are drawn at random without replacement. Using E(S_n) = Σ [1 − C(N − N_i, n) / C(N, n)], where N_i is the count of species i, the expected number of species in the subsample of n = 2 is ____ (give the answer to one decimal place).
Numerical answer — type the value.
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Answer: 1.6
C(10, 2) = 45. For the species with 6 individuals, C(4, 2) = 6, so 1 − 6/45 = 0.8667. For 3 individuals, C(7, 2) = 21, so 1 − 21/45 = 0.5333. For 1 individual, C(9, 2) = 36, so 1 − 36/45 = 0.2. The sum is 1.6. Check: the chance that both drawn individuals share a species is (15 + 3 + 0)/45 = 0.4, so E(S_2) = 1 × 0.4 + 2 × 0.6 = 1.6.In a brachiopod population two specimens are measured: x = 4 mm and y = 10 mm for the smaller, and x = 64 mm and y = 40 mm for the larger. Assume y = a x^b holds exactly. The predicted value of y, in mm, for a specimen with x = 100 mm is ____.
Numerical answer — type the value.
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Answer: 50
The slope on a log–log plot is b = ln(40/10) / ln(64/4) = ln 4 / ln 16 = 0.5, which is below the isometric value of 1 for two lengths, so this is negative allometry. Then y(100) = 10 × (100/4)^0.5 = 10 × 5 = 50 mm. Using a straight-line (linear) extrapolation instead of the power law is the usual slip.The lengths of n = 25 brachiopod shells have a mean of 30.0 mm and a standard deviation of 5.0 mm. For a 95% confidence interval of the mean use x̄ ± t s / √n with t = 2.064 (24 degrees of freedom). The full width of the interval, from lower to upper limit, in mm, is ____ (give the answer to two decimal places).
Numerical answer — type the value.
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Answer: 4.13
The standard error is s / √n = 5.0 / 5 = 1.0 mm. The half-width is t × SE = 2.064 mm, so the interval is 27.94 to 32.06 mm and its full width is 2 × 2.064 = 4.128, which is 4.13 mm. Reporting only the half-width (2.06) is the usual slip.