Equations and formulas sheet Chi Square Statistics Equations and formulas sheet We know how to calculate the expected ratio of certain events; calculate the probabilities (rule on mult and add) But is it likely that when we actually observe the data we will see the expected numbers? No, chance may throw off those numbers What is chance? Hard to explain but we know it…(?) If I flip a coin 3 times in a row, possible to get 3 heads? Sure What about 4 times 5X 6x when is something up??

51 SB Expect 26/25 split (50%/50%) 27T/24H (53%/47%) 4 previous tails/this year heads At what point would you start suspecting a setup?

In an experiment we often start by making a hypothesis – this hypothesis allows one to predict/explain a result If the result is numeric we have an expected outcome and an observed outcome Super Bowl Coin Flips Exp = 26/25 Obs = 27/24 Chi-square statistics tells us if the variation we see between expected data and observed data is due to chance, or is too great for chance If difference is due to chance then the expected and observed results are essentially the same If difference is too great for chance then some unknown factor is influencing results

This relates to our inherent hypothesis (alternative vs null**) The coin is fair null = there is no difference between expected and observed The coin is not fair alternative = there is a difference between expected and observed If variation is only due to chance = in this case accept the null hypothesis there is no difference between expected and observed If too much variation to explain by chance = in this case accept the alternative hypothesis the expected and observed are not the same *Null = there is no difference in the outcome of the control and experimental groups (the IV does not influence the DV) …or there is no difference between the expected and observed results *Alternative = there is a difference

This gets complicated… Take Mendel’s dihybrid cross 9:3:3:1 Expected = 1575 for “9” Observed = 1530 Are they the same? So we’re going to start with a simple hypothesis I’m very upset with the Mars Company I’m not getting enough blue M and M’s I don’t think they really put in 24% blue M & M’s The M and M company accurately reports their color distribution Null = there will be no difference between the expected and observed numbers The M and M company does not accurately report their color distribution Alternative = there will be a difference between the expected and observed

Lab 11 Chi-Square Analysis Title lab and date left hand margin (are you keeping up with table of contents)? Copy the following objective Objective Students will understand the goal of the chi-square test and how to apply it. Published distribution of M & M colors 24% blue 14% brown 16% green 20% orange 13% red 14% yellow

Steps to Chi-square analysis Establish hypotheses Null = there is no difference between the data sets Alternative = there is a difference between the data sets 2. Obtain/calculate expected and observed values (can’t use % directly) 3. Use expected and observed values and equation above to determine chi square value (X2) 4. Determine degrees of freedom for the data set using equation df = n - 1 5. Using the degrees of freedom and the chi square distribution table, determine the critical value (CV) 6. Compare chi square value (3 above) to critical value (5 above) If X2 < CV data sets are the same “fail to reject null hypothesis” If X2 > CV data sets are different “reject null hypothesis”

3. Use expected and observed values and equation above to determine chi square value (X2) The smaller X2 the closer the expected and observed But how small does it have to be to accept the null hypothesis? That’s where critical value comes in… 4. Determine degrees of freedom for the data set using equation df = n – 1 Degrees of freedom = different possible results - 1 5. Using the degrees of freedom and the chi square distribution table, determine the critical value (CV) On CSDT find CV with your df and probability value of p = 0.05 This CV is saying “If your CSV is smaller than this CV, there’s a 95% chance expected and observed are the same” 95% because… Sometimes exp and obs might be very far apart by chance, but really be the same Exp and obs might be very close by chance, but really be different “fail to reject” that they are the same (might be same by chance)

A card dealing machine is supposed to deal cards at random, as if from an infinite deck. In a test of the machine (which shuffles and deals cards), the machine was loaded with 50 complete decks of only suit cards (52 cards per deck) and dealt the following before the machine was stopped: Spades 404 Hearts 420 Diamonds 400 Clubs 376 Is the machine dealing the suits randomly?

Results of chi-square analysis have to be considered in light of original hypotheses Controlled experiment Are results of CG same/diff than EG? Do more cancer cells die in control group (without drug) versus experimental group (with drug)? Hypothesis testing experiment Are the observed results the same as hypothesized (expected) results? Is observed inheritance of pea plant traits consistent with hypothesis of method of inheritance? Observational study Is what is observed in one group different from what is observed in another group? Do bird survivors of a drought have deeper beak depth than nonsurvivors?