C. Opinions of several individuals whose names you obtained by sticking a pin through a phone book, and calling the “pinned” name on each page D. Opinions of an ardent Democrat and an ardent Republican E. Today’s price in Australian dollars of the Euro and the Japanese yen. Exercise 2.28. On the basis of the results in the following contingency tables, would you say that sex and survival are independent of one another in Table A? In Table B? CHAPTER 2 PROBABILITY 59 Table A Alive Dead Men 15 5 Women 15 10 Table B Alive Dead Men 15 10 Women 15 8 Exercise 2.29. Provide an example in which an observation X is indepen- dent of the value taken by an observation Y, X is independent of a third observation Z, and Y is independent of Z, but X, Y, and Z are dependent. 2.5. APPLICATIONS TO GENETICS All the information needed to construct an organism, whether a pea plant, a jellyfish, or a person, is encoded in its genes. Each gene contains the information needed to construct a single protein. Each of our cells has two copies of each gene, one obtained from our mother and one from our father. We will contribute just one of these copies to each of our offspring. Whether it is the copy we got from our father or the one from our mother is determined entirely by chance. You could think of this as flipping a coin: One side says “mother’s gene,” the other side says “father’s gene.” Each time a sperm is created in our testis or an ovum in our ovary, the coin is flipped. There may be many forms of a single gene; each such form is called an allele. Some alleles are defective, incapable of constructing the necessary protein. For example, my mother was rh-, meaning that both her copies of the rh gene were incapable of manufacturing the rh protein that is found in red blood cells. This also means that the copy of the rh gene I obtained from my mother was rh But my blood tests positive for the rh protein, which means that the rh gene I got from my father was rh+. Exercise 2.30. The mother of my children was also rh What proportion of our children would you expect to be rh-? Exercise 2.31. Sixteen percent of the population of the United States are rh What percentage do you expect to have at least one rh- gene? (Remember, as long as a person has even one rh+ gene, they can manufac- ture the rh protein.) The gene responsible for making the A and B blood proteins has three alleles, A, B, and O. A person with two type O alleles will have blood type O. A person with one A allele and one B allele will have blood type AB. Only 4% of the population of the United States have this latter blood type. Our genes are located on chromosomes. The chromosomes come in pairs, one member of each pair being inherited from the father and one from the mother. Your chromosomes are passed onto the offspring inde- pendently of one another. Exercise 2. 32. The ABO and rh genes are located on different chromo- somes. What percentage of the population of the United States would you expect to have the AB rh+ blood type? Exercise 2. 33. Forty-five percent of the population of the United States have type O blood. That is, they do not test positive for either the A or the B protein. What percentage of the population do you expect to have at least one O allele? 2.6. SUMMARY AND REVIEW In this chapter, we introduced the basics of probability theory and inde- pendence, considered the properties of a discrete probability distribution, the binomial, and applied the elements of probability to genetics. We learned how to use the BoxSampler add-in to generate random samples from various distributions. Exercise 2.34. Make a list of all the italicized terms in this chapter. Provide a definition for each one, along with an example. Exercise 2.35. (Read and reread carefully before even attempting an answer.) A magician has three decks of cards, one with only red cards, one that is a normal deck, and one with only black cards. He walks into an adjoining room and returns with only a single deck. He removes the top card from the deck and shows it to you. The card is black. What is the probability that the deck from which the card came consists only of black cards? 60 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® Exercise 2.36. An integer number is chosen at random. What is the probability that it is divisible by 2? What is the probability that it is divisi- ble by 17? What is the probability that it is divisible by 2 and 17? What is the probability that it is divisible by 2 or 17? (Hint: A Venn diagram would be a big aid in solving this last part.) Exercise 2.37. Pete, Phil, and Myron are locked in a squash court after hours with only a Twinkie and a coin between them. The only thing all three can agree on is that they want a whole Twinkie or nothing. Myron suggests that Pete and Phil flip the coin, and that the winner flips a coin with him to see who gets the Twinkie. Phil, who is a graduate student in statistics, says this is unfair. Is it unfair and why? How would you decide who gets the Twinkie? CHAPTER 2 PROBABILITY 61 [...]... 110 34 22 12 349 2 121 118 34 110 22 12 273 3 121 110 34 118 22 12 265 4 118 110 34 121 22 12 262 5 121 118 22 110 34 12 261 6 121 110 22 118 34 12 253 7 121 118 12 110 34 22 251 8 118 110 22 121 34 12 250 9 121 110 12 118 34 22 243 10 118 110 12 121 34 22 240 11 121 34 22 118 110 12 177 12 118 34 22 121 110 12 1 74 13 121 34 12 118 110 22 167 14 110 34 22 121 118 12 166 15 118 34 12 121 110 22 1 64 16... of cells, things seem to even out With small numbers, chance seems to predominate Things got worse, when I went to seed the cells into culture dishes These dishes, made of plastic, had a rectangular grid cut into their 70 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL bottoms, so they were divided into approximately 100 equal-sized squares Dropping 100 cells into the dish meant an... value each time Plot the histogram of these means Compare with the histograms of a) a sample of 512 normally distributed observations with expected value 3.5 and variance 2.3, b) a sample of 512 binomial observations each consisting of 10 trials with probability of success p = 0 .4 per 74 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL FIGURE 3 .4 Preparing to generate the mean values... contamination? Suppose these cultures had escaped contamination and given rise to the observations 90 and 95; what would be the results of a permutation 80 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL analysis applied to the new, enlarged data set consisting of the following cell counts Treated 121 118 110 90 Untreated 95 34 22 12? Hint: To determine how probable an outcome like this is by... E {X + Y} = Â j (EX + j ) Pr {Y = j } = EX *1 + EY 4 In real life, expectations almost always exist and are finite—the expectations of ratios are a notable exception 76 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL A similar result holds if X and Y have continuous distributions, providing that their individual expectations exist and are finite Exercise 3.15 Show that for any variable... situations, we would also want to know the magnitude of the effect Does vitamin E extend cell life span by 3 cell generations? By 10? By 15? In Section 1.6.2 we showed how to use the bootstrap to estimate the precision of the sample mean or median (or, indeed, almost any sample 82 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL statistic) as an estimator of a population parameter As... transplanted the cells, let them grow for 24 hours in contact with a radioactive label, and then fixed and stained them before covering them with a photographic emulsion Ten days passed, and we were ready to examine the autoradiographs “121, 118, 110, 34, 12, 22.” I read and reread these six numbers over and over again The larger numbers were indicative of more cell generations and an extended life span If the... can determine whether the addition of fertilizer increases the resulting yield of tomatoes, at least as far as these dozen plants are concerned But can we extend our findings to all tomatoes? To ensure that we can extend our findings we need to proceed as follows: First, the 12 tomato plants used in our study have to be a random sample from a nursery If we choose only plants with especially CHAPTER 3 DISTRIBUTIONS... click the double arrow ᭤᭤ on the simulation bar to generate a set of bootstrap results for the difference in means in column R Finally, we use Excel s sort command to sort these differences in descending order In the simulation I ran, the largest differences were 7.02 6.11 4. 70 1. 74 1 .43 1.08 and -0.01 I discarded the top 5 as well as the bottom 5 in value to obtain a 90% (90 out of a 100) confidence interval... equal growth potential The alterna- 78 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL tive hypothesis of interest was that cells grown in the presence of vitamin E would be capable of many more cell divisions Under the null hypothesis, the labels “treated” and “untreated” provide no information about the outcomes: The observations are expected to have more or less the same values in . distribution F than if it has cumulative distribution G. 64 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® –3 –2 –1 0 1 2 3 0.0 0.2 0 .4 0.6 0.8 1.0 values P {X < value} FIGURE 3.1. MICROSOFT OFFICE EXCEL ® FIGURE 3 .4 Preparing to generate the mean values of binomial samples. and If X and Y are independent discrete random variables and their expecta- tions exist and are finite 4 ,. distributed random variables with finite expectation q is also q. 76 STATISTICS THROUGH RESAMPLING METHODS AND MICROSOFT OFFICE EXCEL ® The variance of the sum of two independent variables X and Y is