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CHI SQUARED ANALYSIS OF ECOLOGICAL SAMPLING WORKED EXAMPLE: An area on the summit of Sgùrr Choinnich Mor in Scotland is grazed by sheep in the summer and frequented by hillwalkers. There are areas of the summit containing heather (Canna vulgaris) and some containing moss (Rhytidiadelphus squarrosus). A visual survey of this area suggested that the moss is associated with the heather. We want to know if the two species (heather and moss) are present together due to random chance or if they have a relationship with one another (either a positive or negative influence on each other). Step 1: Write a null (H0) hypothesis and and alternative (H1) hypothesis. Null always predicts a scenario with no association whereas the alternative will always predict a scenario where the two populations are associated. Null hypothesis (H0): Alternative hypothesis (H1): A series of 200 quadrats were set up on the summit and the presence or absence of heather or moss in each quadrat was recorded. Results: Species Heather only Moss only Both species Neither species Frequency (Number of quadrats specie appeared in) 31 45 89 35 Step 2: Construct a contingency table of observed values using the data above. Heather present Moss present Heather absent Row totals 89 Moss absent 31 Column totals 120 200 Step 3: Calculate the expected frequencies, assuming that the species are NOT affecting each other (independent distribution). Assuming that the two species are randomly distributed with respect to each other, the probability of moss and heather being present in a quadrat is: (row total x column total) total number of quadrats = (134 x 120) 200 = 80 (round up) Expected values follow the assumption that totals for each row and column do not change, because the relationship shown by the data is assumed to represent the true relative frequency of each species Heather present Heather absent Row total Observed 89 Expected 80 Observed 31 Expected 40 Moss present Moss absent Column total 120 Step 4: Calculate chi squared using the following equation: 200 The chi squared in this example can be calculated as follows: = =____________________________ Step 5: Calculate the degrees of freedom using this equation and the # of rows and columns in the contingency table. Degrees of freedom = (number of rows – 1) x (number of columns – 1) =_______________________ Step 6: Find the critical value at a 95% confidence limit and circle it Step 7: Is your chi squared value LARGER than the critical value? __________ Then you reject your null hypothesis and accept your alternate hypothesis There IS a statistically significant association between moss and heather in this area on the summit of Sgùrr Choinnich Mor. The distribution of the two species are not independent of each other, the distribution of the two species is associated. CHI SQUARED M&M INVESTIATION One researcher has suggested that _____________ colour M&Ms have a negative effect on _____________ M&Ms. The researcher suggested that _____________ M&Ms outcompete _____________ M&Ms for resources. Step 1: Write a null (H0) hypothesis and and alternative (H1) hypothesis. Null always predicts a scenario with no association whereas the alternative will always predict a scenario where the two populations are associated. Null hypothesis (H0): Alternative hypothesis (H1): A series of 20 quadrats were set up on the desk and the presence or absence of _____________ colour M&Ms or _____________ colour M&Ms in each quadrat was recorded. Results: Species _____________ only _____________ only Both species Neither species Frequency (Number of quadrats specie appeared in) Step 2: Construct a contingency table of observed values using the data above. _____________ present _____________ absent Row totals _____________ present _____________ absent Column totals Step 3: Calculate the expected frequencies, assuming that the species are NOT affecting each other (independent distribution). _____________ present _____________ present _____________ absent Observed Expected Observed Expected Column total Step 4: Calculate chi squared using the following equation: The chi squared in this example can be calculated as follows: = = =____________________________ _____________ absent Row total Step 5: Calculate the degrees of freedom using this equation and the # of rows and columns in the contingency table. Degrees of freedom = (number of rows – 1) x (number of columns – 1) =_______________________ Step 6: Find the critical value at a 95% confidence limit and circle it Step 7: Is your chi squared value LARGER or SMALLER than the critical value? What does this mean? __________