**You must type this and all other homework assignments. Do not e-mail the assignment to me; turn it in early (at 322 Wolf) for a foreseeable absence, or turn it in late after an unexpected absence from class. **

**1. **Now we're going to ask the question, does the sex of the person observing balance time have a relationship to balance time? First, randomly pick 10 male observers and 10 female observers. Say what method you used to randomly pick the observers. Analyze the data using nested anova, with sex of the observer as the groups, individual observers as the subgroups, and log transformed balance time as the measurement variable. Report the F-statistic, degrees of freedom, and P-value for the groups and subgroups, and report the percentage of variation explained by each level. What do you conclude? Based on this information, how would you design a future set of observations on balance time?

**I picked 10 male and 10 female observers at random using the RANDBETWEEN() function in Excel, by entering "=RANDBETWEEN(2,583)" and copying and pasting this into a column. This gave me a bunch of random numbers from 2 to 583. I went to the row of the first random number (row 91) and copied the 8 rows for one observer that included row 91. I kept doing this for each random number until I had 10 observers of each sex. I skipped observers if I already had them once, or if they did not have 8 subjects. Through some clever use of the "COPY...PASTE SPECIAL...TRANSPOSE" commands, I was able to quickly get the copied data in the correct format for the nested anova spreadsheet.**

**The results were F _{1, 18}=0.003, P=0.957 for groups, and F_{18, 140}=4.57, P=9x10^{-8} for subgroups. The variance components were 0 percent for among groups, 31 percent for among subgroups, and 69 percent within subgroups. Of course, you'll have a different set of random observers, so your results will be different, but I suspect they'd be fairly similar.**

**I conclude that there's no significant difference in balance time between male and female observers, but there's highly significant variation among observers. So some observers have subjects with much higher balance times than other observers. This might be because some observers pick a set of subjects that really have shorter balance times than others, but it seems more likely that some observers get shorter balance times from their subjects due to some differences in their observing technique. If I were designing a "real" study of balance time, I would make sure to carefully train and test observers to make sure that they all used the exact same technique. **

**2. **It seems plausible that if there's an effect of the observer's sex on balance time, there might be an interaction between the sex of the observer and the sex of the balancer. So using the entire data set (not just the ones you analzyed in question 1), randomly pick 40 measurements of log-transformed balance time for each combination of observer sex and balancer sex. Say what method you used to randomly pick the observers. Use this online two-way anova calculator to do a two-way anova. (Don't worry about "weighted" vs. "unweighted" analysis, you're doing a balanced design so the two methods are identical.) Report the three P-values and give your interpretation of the results. If any of the P-values are significant, give your speculation about why there might be the difference you found.

**Using the random numbers I generated with the RANDBETWEEN function for question 1, I sorted the datasheet first by observer sex, then by subject sex, then by random number. I then copied the first 40 numbers for each combination of sexes and pasted them into the two-way anova web page. My results were F=0.18, P=0.67 for the main effect of observer sex, F=3.84, P=0.052 for the main effect of subject sex, and F=2.64, P=0.11 for the interaction. The interaction term is not significant (although it is interestingly kind of low), so we can look at the main effects. There's no significant effect of observer sex, confirming the results from the nested anova. There's an almost-significant effect of subject sex. Looking at the means for each combination, balance time for male subjects is a little longer than for female subjects when the observers are male, but when the observers are female, the balance time for male subjects is a lot longer than for females. Fascinating--further research is needed!**