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Consider the information in Exercise 3.28 on page 93. The

Probability and Statistics for Engineers and the Scientists | 9th Edition | ISBN: 9780321629111 | Authors: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye ISBN: 9780321629111 32

Solution for problem 4.28 Chapter 4

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Probability and Statistics for Engineers and the Scientists | 9th Edition | ISBN: 9780321629111 | Authors: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye

Probability and Statistics for Engineers and the Scientists | 9th Edition

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Problem 4.28

Consider the information in Exercise 3.28 on page 93. The problem deals with the weight in ounces of the product in a cereal box, with f(x)=2 5, 23.75 x 26.25, (a) Plot the density function. (b) Compute the expected value, or mean weight, in ounces. (c) Are you surprised at your answer in (b)? Explain why or why not.

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BSC2010 Exam 3 04/21/2015 ▯ Evolution= the change in biological populations over time  Populations evolve, individuals do not o The fossil record o Laboratory experiments o Natural populations ▯ Evolutionary Theory=set of ideas about how evolution takes place  Unifying principle of biology  Dobzhansky said “nothing in biology makes sense except in the light of evolution” ▯ Evolution is NOT just a theory  Well-supported  Parsimonious (simplest explanation)  Falsifiable  Refers to the totality of our understanding, NOT a single hypothesis ▯ Darwin’s grandfather first proposed that species change  No evidence  Lacked a workable mechanism of evolution ▯ Jean-Baptiste Lamarck  French naturalist  Attempted to explain the mechanisms of evolution before Darwin was even born  Found that unimportant traits become reduced or lost (atrophy)  Organisms inherit traits that their parents acquired ▯ Lamarck’s theory of evolution involved required traits, Darwin’s theory of evolution involved heritable traits ▯ Darwin was influenced by  Charles Lyell (geology)  Thomas Malthus (population dynamics) ▯ Natural Selection  The differential reproductive success of certain individuals with more or less desirable traits  Only acts on heritable traits  Required for natural selection to occur: o Population must exhibit some variation in a trait  Variation results from genetic mutation o The trait that exhibits variation must be heritable  A trait is heritable when it can be passed on to one’s offspring o The trait must impact an individual’s reproductive success  Differential reproductive success means that individuals with more desirable traits will be more likely to pass those traits on to future generations ▯ Gene Pool= sum of all alleles in a population  Mutation can introduce a novel allele to the gene pool  Mutations are random ▯ Adaptation= beneficial mutation/trait that is favored by natural selection  Note that natural selection produces adaptations ▯ Heritability (H^2)  H^2=genetic variation/genetic + environmental variation=Vg/Vp  Heritability is the fraction of variability that is genetic  Not the same as “having a genetic basis” o Nearly all traits have a genetic basis  If trait is extremely heritable, then H^2 is 1 o If trait is not heritable, then H^2 is 0  Can be estimated by taking the slope of a parent-offpsring regression line ▯ Reproductive Fitness  Fitness refers to the contribution of genotype/phenotype to the next generation o Ex: slow rabbits will get killed by predators, so the average speed of rabbits will slowly increase  Breeder’s equation: R= H^2 * S o Allows animal and plant breeders to predict the strength of response when they apply artificial selection ▯ Modes of Selection  Stabilizing o Average phenotype is most fit o Mean stays the same o Variation decreases  Ex: birth weight  Disruptive o Both extremes of phenotype re most fit o Mean becomes bimodal o Variation  Ex: island has large seeds and seeds are buried in the ground, so birds with long beaks and short/strong beaks are favored  Directional o One extreme of phenotype is most fit o Mean shifts in one direction  Could be positive or negative o Variation decreases ▯ Frequency Dependent Selection  Positive o Common phenotypes favored  Ex: poisonous type of butterfly has certain wing pattern, that poisonous butterfly’s fitness increases as it becomes more common  So that birds are aware not to eat that type of butterfly  Negative o Rare phenotype o Ex: side-blotched lizards  Red(orange)= territorial  Blue= maintain only 1 female at a time  Yellow= sneaker males o Influenzai  The newest/rarest virus that has the best chance of spreading  Old viruses (which are more common) do not have a good chance of spreading o Floral parts ▯ Sexual Selection= a type of non-random mating in which an organism’s phenotype impacts its chance of mating  Intersexual selection (female choice)  Intrasexual selection (male competition) Intersexual Selection  Driven by female choice  Chosen based on 3 observable traits: o Male may provide female with direct benefits to assist her in survival and reproduction o Indicator of health or longevity (“good genes”) o Sensory bias (looks attractive)  Ex: male guppies developed bright orange tails because females noticed them more  Risk associated: noticeable traits may attract predators o Ex: big feathered peacocks are noticeable to lions ▯ Intrasexual Selection  Male competition o Males may become physical for access to females (ex: fight each other) o Sneaker males  Males remain small so they can avoid larger males ▯ Genetic Drift= unpredictable, random fluctuations in allele frequencies from one generation to another because of a population’s small size  Bottleneck effect o Occurs when a large portion of a population becomes wiped out o Typically from natural disaster o Surviving population is no longer genetically representative of the original population  Founder effect o Genetic drift that occurs when a few individuals become isolated from the rest of the population ▯ Gene Flow  Another word for migration  Exchange of genes from one population to another  Increases genetic variation  Decreases differences between both populations ▯ Hardy Weinberg  Assumptions o Random mating o No mutations o No gene flow (migration) o Population size must be very large (infinite in size-no drift) o No selection (natural, artificial, etc.) ▯ Genotype frequency  # of individuals with the genotype of interest/# of individuals in the population  smaller # over total ▯ Allele Frequency  # of copies of a particular allele in a population/ number of copies of all alleles in the population  smaller # over total ▯ HW Theorem  p^2 +2pq+q^2=1 o p^2= homozygous dominant o 2pq= heterozygote o q^2= homozygous recessive  NO evolution when the equation equals 1 ▯ Create and Maintain Variation  Mutation  Migration (gene flow)  Disruptive or negative frequency-dependent selection  Heterozygote advantage ▯ Nucleotide Substitution  Synonymous substitution  Nonsynonymous substitution ▯ Neutral Theory  Explains that most molecular variations in most populations are selectively neutral (don’t convey a selective advantage or disadvantage)  M=2Nu(1/2N) o Ex: gene pool 1/2N=1/2(20)=1/40 o Rate of fixation of neutral mutations is independent of population size (because 2N cancels out) o Rate of fixation of neutral mutations=mutation rate ▯ Purifying (Negative Selection)  Synonymous mutations>Nonsynonymous mutations (more to less) o Nature has been removing mutations (most likely deleterious) o No real change is happening to amino acid sequence (stabilizing selection) ▯ Directional (Positive Selection)  Nonsynonymous mutations>Synonymous mutations (more to less) o Nature is keeping non-neutral mutations o Changes amino acid sequence ▯ Sexual Reproduction  Advantages o Recombination- increases variation in offspring o Repairs damaged DNA (you have 2 copies) o Defense against diseases  Disadvantages o Recombination breaks up good genes o Reduced rate of reproduction ▯ Asexual Reproduction  Advantages o Higher reproductive rate o No need for physical contact to reproduce  Disadvantages o No variation o Muller’s ratchet  Cant get rid of a bad mutation ▯ Lateral Gene Transfer  Swapping DNA from one cell to another o Transduction  Virus inputs DNA into host o Transformation  Prokaryotic cells take up DNA from environment o Conjugaion  Two cells link up and exchange DNA o Hybridization  Exchange of genes between 2 different species ▯ Species Concepts  Morphological (Linneaus) o Based on observable characteristics o Species are groups that look familiar o Limitations:  Members of the same species may not look alike (look different)  Ex: male and female wood ducks look different  Members of different species may look similar  Ex: cryptic species  Biological (Mayr) o Species are groups that can actually or potentially mate with each other o Limitations:  Asexual organisms can’t mate  Extinct species/fossils can’t mate  Hybrids (ex. Horse+donkey=mule, but horses and donkeys are not the same species)  Ring species  Lineage (Simpson) o Species are groups that share a branch on the tree of life o We create phylogenies based on relationships between species o Relies a lot on DNA sequencing o Limitations:  2 species that look the same and interbreed may differ in only neutral mutations ▯ Dobzhansky-Muller Model= explains how a single lineage can split into 2 reproductively isolated species  Applies to major chromosomal rearrangements  ▯ ▯

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Chapter 4, Problem 4.28 is Solved
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Textbook: Probability and Statistics for Engineers and the Scientists
Edition: 9
Author: Ronald E. Walpole; Raymond H. Myers; Sharon L. Myers; Keying E. Ye
ISBN: 9780321629111

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Consider the information in Exercise 3.28 on page 93. The