# LAB : THE CHI-SQUARE TEST. Probability, Random Chance, and Genetics

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1 Period Date LAB : THE CHI-SQUARE TEST Probability, Random Chance, and Genetics Why do we study random chance and probability at the beginning of a unit on genetics? Genetics is the study of inheritance, but it is also a study of probability. Most eukaryotic organisms are diploid, meaning that each cell contains two copies of every chromosome, so there are two copies of each gene that controls a trait (alleles). In sexual reproduction, these two copies of each chromosome separate, and are randomly sorted into the reproductive cells (gametes). When gametes from two different parents combine in fertilization, new combinations of alleles are created. Thus chance plays a major role in determining which alleles, and therefore which combinations of traits end up in each new individual. The important point is that the inheritance of characteristics is the result of random chance. Therefore, it is important to understand the nature of chance and probability and the resulting implications for the science of genetics. In short, the genes that an individual organism inherits depends on the luck of the draw, and the luck of the draw is dependent on the laws of probability. The Laws of Probability There are three Laws of Probability that are important in genetics and they can be easily demonstrated using simple models like flipping a coin or choosing cards from a deck: The Rule of Independent Events: Past events have no influence on future events. Question: If a coin is tossed 5 times, and each time a head appears, then what is the chance that the next toss will be heads? Answer: 1/2 (1 chance in 2), because coins have 2 sides. The Rule of Multiplication: The chance that two or more independent events will occur together is equal to the product of the probabilities of each individual event. Question: What are the chances of drawing a red nine from a standard deck of cards? Answer: 1/26 (1 chance in 26), because there is 1/2 chance of drawing a red card and 1 chance in 13 of drawing a nine. Therefore, 1/2 x 1/13 = 1/26 or 1 chance in 26 of drawing a red nine. The Rule of Addition: The chance of an event occurring when that event can occur two or more different ways is equal to the sum of the probabilities of each individual event Question: If 2 coins are tossed, what is the chance that the toss will yield 2 unmatched coins (1 head & 1 tail)? Answer: 1/2 (1 chance in 2) because the combination of 2 unmatched coins can come about in 2 ways: Result A (coin #1 heads, coin #2 tails) as well as Result B (coin #1 tails, coin #2 heads). Therefore (1/2 x 1/2) + (1/2 x 1/2) = 1/2, or the chance of Result A plus the chance of Result B. 1 of 13

6 nothing else but random chance is at work here. So, as a scientist, you would state your "acceptable" results from the Chi-square analysis in this way: "The differences observed in the data were not statistically significant at the.05 level." You could then add a statement like, "Therefore the data support the hypothesis that..." And you will see that over and over again in the conclusions of research papers. This is how a scientist would state "unacceptable" results from the Chi-square analysis: "The differences observed in the data were statistically significant at the.05 level." You could then add a statement like, "Therefore the data do not support the hypothesis that..." N.B.: Do not forget in your writings that the word "data" is plural (datum is singular & rarely used). "Data are" is correct. "Data is" is not correct. 6 of 13

7 TEAM DATA Toss H/H H/T T/T Toss H/H H/T T/T Toss H/H H/T T/T Total* * The sum of the total of each column must equal 100 tosses. 7 of 13

8 CLASS DATA H/H H/T T/T Totals Obs Exp Total TEAM DATA: CHI SQUARE ANALYSIS H/H H/T A B C D E Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 Exp T/T X 2 Total Degrees of Freedom CLASS DATA: CHI SQUARE ANALYSIS H/H H/T A B C D E Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 Exp T/T X 2 Total Degrees of Freedom 8 of 13

9 CHI-SQUARE DISTRIBUTION TABLE Degrees of freedom Probability (p) value ACCEPT NULL HYPOTHESIS REJECT In scientific research, the probability value of 0.05 is taken as the common cut off level of significance. A probability value (p-value) of.05 means that there is a 5% chance that the difference between the observed and the expected data is a random difference, and a 95% chance that the difference is real and repeatable in other words, a significant difference. Therefore, if your p-value is greater than.05, you would accept the null hypothesis: The difference between my observed results and my expected results are due to random chance alone and are not significant. In genetics experiments (like your upcoming Fly Lab), accepting the null hypothesis would mean that your data are supporting your proposal for the genetics and inheritance scheme for the flies that you were breeding. In medical research, the chi-square test is used in a similar but interestingly different way. When a scientist is testing a new drug, the experiment is set up so that the control group receives a placebo and the experimental group receives the new drug. Analysis of the data is trying to see if there is a difference between the two groups. The expected values would be that the same number of people get better in the two groups which would mean that the drug has no effect. If the chi-square test yields a p-value greater than.05, then the scientist would accept the null hypothesis which would mean the drug has no significant effect. The differences between the expected and the observed data could be due to random chance alone. If the chisquare test yields a p-value.05, then the scientist would reject the null hypothesis which would mean the drug has a significant effect. The differences between the expected and the observed data could not be due to random chance alone and can be assumed to have come from the drug treatment. In fact, chi-square analysis tables can go to much lower p-values than the one above they could have p-values of.001 (1 in 1000 chance),.0001 (1 in 10,000 chance), and so forth. For example, a p-value of.0001 would mean that there would only be a 1 in 10,000 chance that the differences between the expected and the observed data were due to random chance alone, whereas there is a 99.99% chance that the difference is really caused by the treatment. These results would be considered highly significant. 9 of 13

10 QUESTIONS 1. What is the Chi-square test used for? 2. Why is probability important in genetics? 3. Briefly describe how the Chi-square analysis may be used in genetics. 4. Suppose you were to obtain a Chi-square value of 7.82 or greater in your data analysis (with 2 degrees of freedom). What would this indicate? 5. Suppose you were to obtain a Chi-square value of 4.60 or lower in your data analysis (with 2 degrees of freedom). What would this indicate? 6. A heterozygous white-fruited squash plant is crossed with a yellow-fruited plant, yielding 200 seeds. Of these, 110 produce white-fruited plants while only 90 produce yellow-fruited plants. Are these results statistically significant? Explain using Chi-square analysis. 10 of 13

12 10. When scientists design research studies they purposely choose large sample sizes. Work through these scenarios to see why: a. Just as in your experiment, you flipped 2 coins, but you only did it 10 times. You collected these data below. Use the chart to calculate the Chi-square value: H/H 1 H/T 8 T/T 1 Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 X 2 Total Exp Would you accept or reject the null hypothesis? Explain using your Chi-square analysis. b. Now you flipped your 2 coins again, but you did it 100 times. You collected these data below. Use the chart to calculate the Chi-square value: H/H 10 H/T 80 T/T 10 Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 X 2 Total Exp Would you accept or reject the null hypothesis? Explain using your Chi-square analysis. 12 of 13

13 c. Now you flipped your 2 coins again, but you did it 1000 times. You collected these data below. Use the chart to calculate the Chi-square value: H/H 100 H/T 800 T/T 100 Obs Exp Obs - Exp (Obs Exp) 2 (Obs Exp) 2 X 2 Total Exp Would you accept or reject the null hypothesis? Explain using your Chi-square analysis. d. Now, using your understanding of the Chi-square test, explain why scientists purposely choose large sample sizes when they design research studies. 13 of 13

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