Let's think more about this third factor: the role of variation in influencing the likelihood of our sample means being different due to chance. Consider Figure 2. Populal'lons Ho: u t -p2 Number uf individuals I .I' \"H... 30 50 . i1 22 Number uf Samples Individual: ' \\ I \\ 30 40 50 30 40 50 Figure 2. Frequency plots showing random samples (bottom rows) taken from populations (upper graphs) with large standard deviations (left) and small standard deviations (right). . The top left graph shows frequency plots of a trait measurement (e. g. time termites spent on a line) for two treatments (e. g. solid lines and dotted lines) in the case where the null hypothesis is true (e.g. mean time on solid lines u1=mean time on dotted lines uz ). These populations have a large standard deviation for this particular trait, ranging between 20-60. When two random samples of individuals were taken and measured for this trait under different treatments (lower left graph) the difference between the means for these two samples (gi1) was 45-35=10. The graph on the top right illustrates what the frequency distributions look like if the standard deviation for this trait is much smaller (values range between 35-45). Note, everything else is the same as on the left (population means ( in and 1.12 still: 40). If you measured two random samples from these populations (lower graph), would you be as likely to get the pictured difference between means (iii] =10), as you were in the case on the left where the standard deviations were larger? What do you think? EXERCISE 2 - Statistically significant difference between means: the t-test Be sure to refer to pages 7-12 in the manual as you work through the questions in this exercise. The top panel in Figure 2 (pg 8) of your lab text has frequency plots showing the number of individuals with each value for some measured variable in large populations. On the left the standard deviations for this variable are larger than the standard deviations of the populations on the right. Two samples were drawn from each large population (bottom panel). In both cases the differences between the sample means (X1-X2) are the same (=10). Are you as likely to draw samples with a difference in mean of 10) from the populations on the right as you are from the populations on the left? What do you think? Why? We If you did draw samples with a mean difference of 10 from the populations on the right, what might you conclude about Hi and u2 of these populations? BBio180, University of Washington, Bothell (Revised Fall 22)