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This exercise continues the study of the robustness of the

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi ISBN: 9780073401331 38

Solution for problem 8E Chapter 5.10

Statistics for Engineers and Scientists | 4th Edition

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Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

Statistics for Engineers and Scientists | 4th Edition

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Problem 8E

Problem 8E

This exercise continues the study of the robustness of the Student's t method for constructing confidence intervals. The following figure shows graphs of probability density functions for the N (0, 1) distribution, the lognormal distribution with µ = 1 and σ2 = 0.25, and the gamma distribution with r = 0.5 and λ = 0.5 (this is also known as the chi-square distribution with one degree of freedom). For each of these distributions, generate 10,000 samples of size 5, and for each sample compute the upper and lower limits of a 95% confidence interval using the Student's t method. [If necessary, it is possible to compute the lognormal and gamma random values from normal random values. Specifically, to compute a value X from a lognormal distribution with µ = 1 and σ2 = 0.25, generate Y ~ N (l, 0.25) and compute X = eY. To generate a value X from a gamma distribution with r = 0.5 and λ = 0.5, generate Y ~ N (0, 1) and compute X = Y2.]

a. The true mean of the N (0, 1) distribution is 0. Based on the simulation results, estimate the coverage probability (proportion of samples for which the confidence interval covers the true mean) for samples of size 5 from the N (0, 1) distribution. (Since the assumptions underlying the Student's t method are satisfied here, your answer should be very close to 95%.)

b. The true mean of the lognormal distribution with µ = 1 and σ2 = 0.25 is 3.0802. Based on the simulation results, estimate the coverage probability (proportion of samples for which the confidence interval covers the true mean) for samples of size 5 from the lognormal distribution with µ = 1 and σ2 = 0.25.

c. The true mean of the gamma distribution with r = 0.5 and λ = 0.5 is 1. Based on the simulation results, estimate the coverage probability (proportion of samples for which the confidence interval covers the true mean) for samples of size 5 from the gamma distribution with r = 0.5 and λ = 0.5.

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Chapter 20 Notes: Evolution of Angiosperms -­‐ Hypothesized that link exists between seed-­‐bearing organs of Mesozoic seed ferns and carpels of carpels of angiosperms -­‐ Anthophyte hypothesis: gnetophytes are closest relatives to angiosperms (based on morphological characters) -­‐ Mystery still remains about angiosperm evolution -­‐ Anthophyta (angiosperms) derived from single common ancestor o Flowers o Seeds enclosed by carpel o Double fertilization leading to endosperm formation o Reduced 3-­‐nucleate microgametophyte o Reduced megagametophyte o Stamens with two pairs of pollen sacs o Presence of sieve-­‐tube elements and companion cells in phloem -­‐ Earliest angiosperm: Archaefructus Early Cretaceous ~125 mya o Small herbaceous aquatic plant with non-­‐showy flowers, lack perianth o Predate appearance of plants with magnolia-­‐like flowers by at least 10 to 20 million years -­‐ Earliest angiosperms produced pollen with single aperture (ancestral condition) -­‐ Monocots had common ancestor, indicated by single cotyledon and other features -­‐ Eudicots also had common ancestor, triaperturate pollen (derived) -­‐ Several evolutionary lines of angiosperms arose before the split between monocots and eudicots -­‐ Basal grade angiosperms: Amborella trichopoda, Nymphaeales, Austrobaileyales; sister groups to other flowering plants o Amborella: shrubby plant, small flowers, lack distinct petals and sepals, imperfect (unisexual, but evolved from bisexual), lack vessels only tracheids o Nymphaeales: herbaceous, aquatic plants adapted to high light intensity; lack vessels o Austrobaileyales: shrubs or small trees adapted to low light intensity; moist tropical forest understory -­‐ Magnoliids: first lineage to diverge with Mesangiospermae -­‐ Monocots: second lineage of mesangiosperms that retain some of basal angiosperm features such as monoaperturate pollen and 3-­‐merous flowers -­‐ Eudicots: third and final lineage of mesangiosperms -­‐ The perianth of early angiosperms did not have distinct sepals and petals o Either petals and sepals were identical or gradual transition in appearance between two whorls o Petals probably derived originally from stamens that lost sporangia o Petal fusion occurred number of times during evolution -­‐ Stamens of early angiosperms were diverse in structure and function o Woody Magnoliids: broad, colored, scented o Other archaic angiosperms: small, greenish, fleshy o Monocots and eudicots: thin filaments and thick, terminal anthers o In some plant families: stamens became secondarily sterile; lost sporangia and become transformed into nectaries: glands that secrete nectary -­‐ Carpels of early angiosperms were unspecialized o Some archaic angiosperms have somewhat leaf-­‐like carpels, with no specialized areas o Carpels either free from each other or incompletely closed o Variation in arrangement of ovules among eudicots 1. From flowers with few to many parts that are indefinite in number, flowers have evolved toward having few parts that are definite in number 2. The floral axis has become shortened so that the original spiral arrangement of parts is no longer evident, and the floral parts often have become fused 3. The ovary has become inferior rather than superior in position, and the perianth has become differentiated into a distinct calyx and corolla 4. The radial symmetry (regularity), or actinomorphy, of early flowers has given way to bilateral symmetry (irregularity), or zygomorphy, in more advanced ones -­‐ Asteraceae (eudicots) and Orchidaceae (monocots) are the most specialized flowers and two largest families of angiosperms -­‐ The flowers of Asteraceae are closely bunched into a head o Each of tiny flowers has inferior ovary composed of two fused carpels with a single ovule in one locule o In composite flowers, stamens reduced to five in number and usually fused to one another and to corolla. Petals fused to one another and to ovary; sepals absent or reduced to pappus (aid to dispersal by wind) o Each head includes two types of flower § Disk flower: make up central portion of aggregate § Ray flower: arranged on outer periphery; often carpellate, sometimes completely sterile -­‐ Orchidaceae is largest angiosperm family o At least 24,000 species of orchids; individual species of orchids are rarely very abundant o Most species tropical o Three carpels fused and, in composites, ovary is inferior; each orchid ovary contains many thousands of minute ovules o Column: stem fused with style and stigma o Pollinium: unit of entire contents of anther held together o Three petals modified so two laterals form wings and third forms cup-­‐ like lip; bilaterally symmetrical o Several genera lack chlorophyll and survive as myco-­‐heterotrophs -­‐ Animals serve as primary agents of floral evolution o Flowers and insects have coevolved § Insects feeding on pollen of other flower parts began returning to these new-­‐found sources of food and thus transferred pollen from plant to plant § More efficient, more accurate pollination § Evolution of closed carpel gave certain seed plants reproductive advantage from being eaten by insects § Bisexual flower offers selective advantage making each visit by pollinator more effective § Selection favors specializations related to characteristics of visitors o Bees most important flower-­‐visiting insects § Have adaptations that make them suitable for collecting and carrying nectar and pollen o “food deception” and “sexual deception” -­‐ Bird and bat pollinated flowers produce copious nectar o Bird-­‐pollinated flowers generally have copious, thin nectar but usually have little odor; flowers usually are colorful, red and yellow o Most flowers pollinated by bats have copious nectar and are dull colored, many opening only at night -­‐ Wind pollinated flowers produce no nectar o Dull colors, relatively odorless; petals either small or absent; sexes often separated on same plant; temperate regions; well-­‐exposed anthers; large stigmas -­‐ Most important pigments in floral coloration are the flavonoids o The way in which pigments are concentrated in flowers, and particularly in corollas, is special characteristic of flowering plants o Flavonoids: compounds with two six-­‐carbon rings linked by three-­‐ carbon unit; block UV radiation; usually selectively admit light of blue-­‐ green and red wavelengths o Anthocyanins: major class of flavonoids; most red and blue plant pigments; water-­‐soluble and found in vacuoles § Color depends on acidity of cell sap of vacuole o Carotenoids: flavonoids that are oil-­‐soluble and found in plastids o Flavonols: commonly found in leaves and also in many flowers; many colorless but may also contribute ivory or white hues o Different mixtures of flavonoids and carotenoids and differences in structural properties of flower parts produce characteristic colors o Betacyanins: reddish pigments; more complex aromatic compounds in goosefoot, cactus, and portulaca families -­‐ Fruit is matured ovary, along with accessory tissue. o Fruits may develop without fertilization and seed development: parthenocarpy; parthenocarpic fruits (banana, citrus, pumpkin, fig, pineapple) o Simple fruits: develop from single carpel or from two or more united carpels (bean pod, cherry, tomato) § When ripe, may be fleshy or dry § Berries: fleshy fruits with one to many seeds; pulpy except exocarp § Drupes: generally one-­‐seeded, usually thin and skin-­‐like exocarp, fleshy mesocarp, stony endocarp, encloses seed § Pomes: develop from compound inferior ovary; noncarpellary tissue and endocarp enclosing seed is cartilaginous § Dehiscent fruits: split open at maturity and commonly contain several seeds • Follicle: derived from single carpel that splits along one side at maturity (legume splits along both sides) • Silique: fruit of mustard family, at maturity two halves split away from central partition • Capsule: derived from compound ovary; capsules released in variety of ways § Indehiscent fruits: do not split open at maturity and usually originate from ovary in which only one seed develops • Achene: small, one seeded fruit; seed attached to pericarp at one point only; pericarp readily separated from seed coat • Samaras: winged achenes • Cypsela: achene-­‐like fruit derived from inferior ovary • Caryopsis, or grain: achene-­‐like fruit that occurs in grasses; seed coat fused to pericarp over entire surface • Nuts: pericarp that is hard or stony throughout; develops from compound ovary with only one functional carpel; generally one seeded • Schizocarp: in parsley family and maples, splits at maturity into two or more one-­‐seeded portions o Aggregate fruits: formed from gynoecium – apocarpous gynoecium – in which carpel retains its identity in mature state (magnolias, raspberries, strawberries) o Fruitlets: individual matured carpels, or ovaries o Multiple fruits: derived from inflorescence o Accessory fruit: any fruit that contains accessory tissue -­‐ Many plants have wind-­‐borne fruits and seeds o Some fruits have wings, formed from perianth parts o Some plants, seed itself, rather than fruit, bears wing or plume o In willows and poplars, seed covered with woolly hairs o In tumbleweeds, whole plant blown along by wind o Other plants shoot seeds aloft: valves of capsules separate suddenly, throwing seeds o Some seeds or fruits of many plants simply drop to ground and dispersed sporadically -­‐ Fruits and seeds adapted for floating are dispersed by water o Can float either because air trapped in some part of fruit or because fruit contains tissue that includes large air spaces -­‐ Fruits and seeds that are fleshy or have adaptations for attachment are dispersed by animals o Majority of fruits in which pericarp is fleshy are eaten by vertebrates; seeds contained by fruits are spread by passing through digestive tract or regurgitated at distance; partial digestion aids germination of seeds by weakening seed coats o When fleshy fruits ripen, undergo series of characteristic changes mediated by hormone ethylene § Rise in sugar content § Softening of fruit caused by breakdown of pectic substances § Change in color from inconspicuous to bright colors o Fruits or seeds may be dispersed by adhering to fur or feathers o Ants important agent in dispersal -­‐ Secondary metabolites, or secondary plant products: include array of chemically unrelated compounds, such as alkaloids, terpenoids, phenolics, quinones, and raphides o Chemicals appear to play major role in restricting palatability or plants that produce them or in causing animals to avoid plants altogether o Same chemicals that act as deterrents to most groups of insect herbivores often act as feeding stimuli for narrowly restricted feeders o Herbivorous insects that are narrowly restricted in their feeding habits to groups of plants with certain secondary plant products are often brightly colored

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Chapter 5.10, Problem 8E is Solved
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Textbook: Statistics for Engineers and Scientists
Edition: 4
Author: William Navidi
ISBN: 9780073401331

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This exercise continues the study of the robustness of the