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The Chi Squared Test Chi-Squared Test • Used to compare data in the field with data in an experiment – Is the difference between what you observed and what you expected due to random chance or some other factor? – Ex. If you toss a coin 10 times, you expect 5 heads and 5 tails, but that might not be what you observe to happen. – There is variability in the real world • Chi-squared enables us to – distinguish the natural variability between what is observed and what is expected – the possibility of something else significantly playing a role Hypothesis Testing • Hypothesis- a causal relationship exists between an underlying factor (variable) and an observable phenomenon. – Ex. Do English ivy leaves grow bigger in the shade? • Relationship- sunlight amount is the variable causing bigger leaves. • Hypothesis testing- Are the effects real? Is there something in this data? – trying to reject a null hypothesis • A statement explaining that there is no causal relationship – Ex. Leaf widths are the same in sunny and shady environments – Alternative to the null hypothesis- the causal relationship exists • Ex. Leaves grow larger in the shade to capture more sunlight • In most cases, experiments do not prove the alternate to the null hypothesis but reject the null hypothesis Critical Value • How much variation should be tolerated before rejecting the null hypothesis? – If the observed deviate from the predictions, how much do we allow to chance? – In biology we usually have a 5% critical value • Known as a probability value, or p-value – Ex. Data is collected on leaf width in shade and sun • A statistical analysis is used – A p-value is generated- if it is less than 5%, we should reject the null » The leaves of shady plants are significantly larger than leaves in the sun Chi-Squared Test • Used in introductory biology to – Test observed outcomes of genetic test compare with Mendel’s predicted outcomes – See how gene frequencies match up to Hardy-Weinberg equilibrium • Usually trying to prove that any variation away from Mendel or Hardy-Weinberg was due to chance – You want to fail to reject the null, p-value is greater than .05 » Proving Mendel and Hardy-Weinberg correct – Can also be used to reject the null (p ≤.05) • Pill bugs choosing one environment over another – Want to show that one environment is desired significantly – In medicine- comparing a drug to a placebo • Null- effects are the same – Chi-squared p-value greater than .05- drug and placebo are equally effective – Chi-squared p-value less than or equal to .05- there is a significant different » Null is rejected Calculating Chi-Squared •O: observed •E: expected •Example- Mendel’s monohybrid cross •Round is dominant over wrinkled •F2 generation should show a 3:1 ratio •Data- 5,474 round, 1,850 wrinkled •This is the observed •Expected data- 5,493 round, 1,831 wrinkled •5474 +1850= 7324 •7324 x .75= 5493 •7324 x .25= 1831 •Is the difference between the observed and the expected significant? Calculating Chi-Squared Degrees of Freedom- one less than the number of results •Two results here, round or wrinkled, so the degrees of freedom is one Calculating Chi-Squared • Once you have determined the value of chisquared, use a Chi-squared table to calculate the p-value: Calculating Chi-Squared • Results: – Our p-value is well above .05 • do not reject the null hypothesis – The observed vs. the expected values are not significantly different » The differences are due to chance • Summary of Steps: 1. Calculate the observed and the expected. 2. Plug into the Chi-squared formula • Get a number answer for χ2 by adding up the last column 3. Calculate degrees of freedom 4. Use a chi-squared table to calculate the p-value • If p≤.05, then reject the null hypothesis Your Turn • Stream snails were marked and recaptured • Question asked: Do snails tend to move upstream or downstream after initial capture? – If no preference, half would go upstream, half would go down stream – Completed in two stream beds: • Sandy stream: 43 of 50 snails were recovered upstream • Rocky stream: 22 of 38 snails were recovered upstream • Do a Chi-squared analysis for each bed type. Answer Answer Rocky bed- do not reject the null hypothesis Sandy bed- definitely reject the null hypothesis