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# Testing Genetic Ratios Help

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## Sampling Theory and Size

The application of statistics to genetic problems can help predict and evaluate outcomes of crosses and experiments.

### Sampling Theory

If we toss a coin, we expect that half of the time it will land heads up and half of the time tails up. This hypothesized probability is based upon an infinite number of coin tossings wherein the effects of chance deviations from 0.5 in favor of either heads or tails cancel one another. All actual experiments, however, involve finite numbers of observations and therefore some deviation from the expected numbers (sampling error) is to be anticipated. Let us assume that there is no difference between the observed results of a coin-tossing experiment and the expected results that cannot be accounted for by chance alone (null hypothesis). How great a deviation from the expected 50 : 50 ratio in a given experiment should be allowed before the null hypothesis is rejected? Conventionally, the null hypothesis in most biological experiments is rejected when the deviation is so large that it could be accounted for by chance less than 5%of the time. Such results are said to be significant. When the null hypothesis is rejected at the 5% level, we take 1 chance in 20 of discarding a valid hypothesis. It must be remembered that statistics can never render absolute proof of the hypothesis, but merely sets limits to our uncertainty. If we wish to be even more certain that the rejection of the hypothesis is warranted we could use the1% level, often called highly significant, in which case the experimenter would be taking only 1 chance in 100 of rejecting a valid hypothesis.

### Sample Size

If our coin-tossing experiment is based on small numbers, we might anticipate relatively large deviations from the expected values to occur quite often by chance alone. However, as the sample size increases, the deviation should become proportionately less, so that in a sample of infinite size the plus and minus chance deviations cancel each other completely to produce the 50 : 50 ratio.

## Degrees of Freedom

Assume a coin is tossed 100 times. We may arbitrarily assign any number of heads from 0 to 100 as appearing in this hypothetical experiment. However, once the number of heads is established, the remainder is tails and must add to 100. In other words, we have n – 1 degrees of freedom(df) in assigning numbers at random to the n classes within an experiment.

EXAMPLE 2.37 In an experiment involving three phenotypes (n = 3), we can fill two of the classes at random, but the number in the third class must constitute the remainder of the total number of individuals observed. Therefore, we have 3 – 1 = 2 degrees of freedom.

EXAMPLE 2.38 A 9 : 3 : 3 : 1 dihybrid ratio has four phenotypes (n = 4). There are 4 – 1 = 3 degrees of freedom. The number of degrees of freedom in these kinds of problems is the number of variables (n) under consideration minus 1. For most genetic problems, the degrees of freedom will be 1 less than the number of phenotypic classes. Obviously, the more variables involved in an experiment the greater the total deviation may be by chance

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