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Tamarin: Principles of Genetics, Seventh Edition
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II. Mendelism and the Chromosomal Theory
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4. Probability and Statistics
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Probability and Statistics
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a normal son (3/4)(1/2) 3/8 a normal daughter (3/4)(1/2) 3/8 an albino son (1/4)(1/2) 1/8 an albino daughter (1/4)(1/2) 1/8 Thus: P 5! (3/8)2(3/8)2(1/8)1(1/8)0 2!2!1!0! 30(3/8)4(1/8)1 30(3)4/(8)5 2,430/32,768 0.074
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of data and reduces them to one or two meaningful values. We examine further some of these terms and concepts in the chapter on quantitative inheritance (chapter 18).
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Hypothesis Testing
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The third way that statistics is valuable to geneticists is in the testing of hypotheses: determining whether to support or reject a hypothesis by comparing the data to the predictions of the hypothesis. This area is the most germane to our current discussion. For example, was the ratio of 787:277 really indicative of a 3:1 ratio Since we know now that we cannot answer with an absolute yes, how can we decide to what level the data support the predicted 3:1 ratio Statisticians would have us proceed as follows. To begin, we need to establish how much variation to expect. We can determine this by calculating a sampling distribution: the frequencies with which various possible events could occur in a particular experiment. For example, if we self-fertilized a heterozygous tall plant, we would expect a 3:1 ratio of tall to dwarf plants among the progeny. (The 3:1 ratio is our hypothesis based on the assumption that height is genetically controlled by one locus with two alleles.) If we looked at the rst four offspring, what is the probability we would see three tall and one dwarf plant We can calculate the answer using the formula for the terms of the binomial expansion: P 4! (3/4)3(1/4)1 3!1! 108 256 0.42
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S TA T I S T I C S
In one of Mendel s experiments, F1 heterozygous pea plants, all tall, were self-fertilized. In the next generation (F2), he recorded 787 tall offspring and 277 dwarf offspring for a ratio of 2.84:1. Mendel saw this as a 3:1 ratio, which supported his proposed rule of inheritance. In fact, is 787:277 roundable to a 3:1 ratio From a brief discussion of probability, we expect some deviation from an exact 3:1 ratio (798:266), but how much of a deviation is acceptable Would 786:278 still support Mendel s rule Would 785:279 support it Would 709:355 (a 2:1 ratio) or 532:532 (a 1:1 ratio) Where do we draw the line It is at this point that the discipline of statistics provides help. We can never speak with certainty about stochastic events. For example, take Mendel s cross. Although a 3:1 ratio is expected on the basis of Mendel s hypothesis, chance could mean that the data yield a 1:1 ratio (532:532), yet the mechanism could be the one that Mendel suggested. In other words, we could ip an honest coin and get ten heads in a row. Conversely, Mendel could have gotten exactly a 3:1 ratio (798:266) in his F2 generation, yet his hypothesis of segregation could have been wrong. The point is that any time we deal with probabilistic events there is some chance that the data will lead us to support a bad hypothesis or reject a good one. Statistics quanti es these chances. We cannot say with certainty that a 2.84:1 ratio represents a 3:1 ratio; we can say, however, that we have a certain degree of con dence in the ratio. Statistics helps us ascertain these con dence limits. Statistics is a branch of probability theory that helps the experimental geneticist in three ways. First, part of statistics deals with experimental design. A bit of thought applied before an experiment may help the investigator design the experiment in the most ef cient way. Although he did not know statistics, Mendel s experimental design was very good. The second way in which statistics is helpful is in summarizing data. Familiar terms such as mean and standard deviation are part of the body of descriptive statistics that takes large masses
Similarly, we can calculate the probability of getting all tall (81/256 0.32), two tall and two dwarf (54/256 0.21), one tall and three dwarf (12/256 0.05), and all dwarf (1/256 0.004) in this rst set of four. Table 4.1 shows this distribution, as well as the distributions for samples of eight and forty progeny. Figure 4.1 shows these distributions in graph form. As sample sizes increase (from four to eight to forty in g. 4.1), the sampling distribution takes on the shape of a smooth curve with a peak at the true ratio of 3:1 (75% tall progeny) that is, there is a high probability of getting very close to the true ratio. However, there is some chance the ratio will be fairly far off, and a very small part of the time our ratio will be very far off. It is important to see that any ratio could arise in a given experiment even though the true ratio is 3:1. At what point do we decide that an experimental result is not indicative of a 3:1 ratio Statisticians have agreed on a convention. When all the frequencies are plotted, as in gure 4.1, we can treat the area under the curve as one unit, and we can draw lines to mark 95% of this area ( g. 4.2). Any ratios in-
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