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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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The McGraw Hill Companies, 2001
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Table 4.1 Sampling Distribution for Sample Sizes of Four, Eight, and Forty, Given a 3:1
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Ratio of Tall and Dwarf Plants
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n No. Tall Plants 4 3 2 1 0 4 Probability* 81 256 108 256 54 256 12 256 1 256 0.32 0.42 0.21 0.05 0.004 No. Tall Plants n 8 Probability* n No. Tall Plants 40 Probability*
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0.10 0.27 0.31 0.21 0.09 0.02 0.004 0.0004 0.00002
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40 39 38 30 2 1 0
0.00001 0.0001 0.0009
0.59 0.10 0.83 10 10 10
20 21 24
* Probabilities are calculated from the binomial theorem. probability (n!/s!t!)psqt where n number of progeny observed s number of progeny that are tall t number of progeny that are dwarf p probability of a progeny plant being tall (3/4) q probability of a progeny plant being dwarf (1/4)
1 Frequency (probability) Frequency (probability) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 25 50 75 100 Percent tall
1 Frequency (probability) 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 25 50 75 100 Percent tall
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 25 50 75 100 Percent tall
n = 40
Sampling distributions from an experiment with an expected ratio of three tall to one dwarf plant. As the sample size, n, gets larger, the distribution curve becomes smoother. These distributions are plotted terms of the binomial expansion (see table 4.1). Note also that as n gets larger, the peak of the curve gets lower because as more points (possible ratios) are squeezed in along the x-axis, the probability of producing any one ratio decreases.
cluded within the 95% limits are considered supportive of (failing to reject) the hypothesis of a 3:1 ratio. Any ratio in the remaining 5% area is considered unacceptable. (Other conventions also exist, such as rejection within the outer 10% or 1% limits; we consider these at the end
of the chapter.) Thus, it is possible to see whether the experimental data support our hypothesis (in this case, the hypothesis of 3:1). One in twenty times (5%) we will make a type I error: We will reject a true hypothesis. (A type II error is failing to reject a false hypothesis.)
Tamarin: Principles of Genetics, Seventh Edition
II. Mendelism and the Chromosomal Theory
4. Probability and Statistics
The McGraw Hill Companies, 2001
Four
Probability and Statistics
To determine whether to reject a hypothesis, we must derive a frequency distribution for each type of experiment. Mendel could have used the distribution shown in gure 4.1 for seed coat or seed color, as long as he was expecting a 3:1 ratio and had a similar sample size. What about independent assortment, which predicts a 9:3:3:1 ratio A geneticist would have to calculate a new sampling distribution based on a 9:3:3:1 ratio and a particular sample size. Statisticians have devised shortcut methods by using standardized probability distributions. Many are in use, such as the t-distribution, binomial distribution, and chi-square distribution. Each is useful for particular kinds of data; geneticists usually use the chi-square distribution to test hypotheses regarding breeding data.
Chi-Square
When sample subjects are distributed among discrete categories such as tall and dwarf plants, geneticists frequently use the chi-square distribution to evaluate data. The formula for converting categorical experimental data to a chi-square value is
(O E
where is the Greek letter chi, O is the observed number for a category, E is the expected number for that category, and means to sum the calculations for all categories. A chi-square ( 2) value of 0.60 is calculated in table 4.2 for Mendel s data on the basis of a 3:1 ratio. If Mendel had originally expected a 1:1 ratio, he would have calculated a chi-square of 244.45 (table 4.3). However, these 2 values have little meaning in themselves: they are not probabilities. We can convert them to probabilities by determining where the chi-square value falls in relation to the area under the chi-square distribution curve. We usually use a chi-square table that contains probabilities that have already been calculated (table 4.4). Before we can use this table, however, we must de ne the concept of degrees of freedom. Reexamination of the chi-square formula and tables 4.2 and 4.3 reveals that each category of data contributes to the total chi-square value, because chi-square is a summed value. We therefore expect the chi-square value to increase as the total number of categories increases. That is, the more categories involved, the larger the chisquare value, even if the sample ts relatively well against the hypothesized ratio. Hence, we need some way of keeping track of categories. We can do this with degrees of freedom, which is basically a count of independent categories.With Mendel s data, the total number of offspring is 1,064, of which 787 had tall stems. Therefore, the
short-stem group had to consist of 277 plants (1,064 787) and isn t an independent category. For our purposes here, degrees of freedom equal the number of categories minus one. Thus, with two phenotypic categories, there is only one degree of freedom. Table 4.4, the table of chi-square probabilities, is read as follows. Degrees of freedom appear in the left column. We are interested in the rst row, where there is one degree of freedom. The numbers across the top of the table are the probabilities. We are interested in the next-to-thelast column, headed by the 0.05. We thus gain the following information from the table: The probability is 0.05 of getting a chi-square value of 3.841 or larger by chance alone, given that the hypothesis is correct. This statement formalizes the information in our discussion of frequency distributions. Hence, we are interested in how large a chi-square value will be found in the 5% unacceptable area of the curve. For Mendel s plant experiment, the critical chi-square (at p 0.05, one degree of freedom) is 3.841. This is the value to which we compare the calculated 2 values (0.60 and 244.45). Since the chi-square value for the 3:1 ratio is 0.60 (table 4.2), which is less than the critical value of 3.841, we do not reject the hypothesis of a 3:1 ratio. But since 2 for the 1:1 ratio (table 4.3) is 244.45, which is greater than the critical value, we reject the hypothesis of a 1:1 ratio. Notice that once we did the chi-square test for the 3:1 ratio and failed to reject the hypothesis, no other statistical tests were needed: Mendel s data are consistent with a 3:1 ratio. A word of warning when using the chi-square: If the expected number in any category is less than ve, the conclusions are not reliable. In that case, you can repeat the experiment to obtain a larger sample size, or you can
Frequency (probability)
0.05 2.5% 2.5%
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