1. Researchers were interested in understanding whether differences in the levels of aggressive behavior seen in the American Tree Sparrow (e.g., all birds of this species) in the summer months were due to differences in levels of testosterone. Of primary interest was whether or not testosterone injections would, on average, result in different levels of aggression. Researchers state that a difference between groups of more than 1 unit on the aggression scale would be scientifically meaningful. A study of sparrows on the Colville National Forest was conducted as follows. Researchers drove to a pre-determined location on the Colville National Forest on July 16th and called a total of 18 birds to a feeder using recorded bird songs. Randomly, nine of the birds were chosen to receive a testosterone injection and all birds were fitted with a radio transmitter. Two weeks later, researchers returned, called the birds again (with the exception of 1 bird which they could not find), and made observations of their behavior. The behavioral observations were summarized as a single aggression score for each bird. The data from this study is contained in the file sparrow.txt in the homework data folder with cont representing the aggression score for the control birds and exp representing the aggression score for the birds that received the testosterone treatment. Read the data into R in order to prepare for analysis. The missing observation in the control group can be disregarded for this analysis. From what you have learned so far in class, use appropriate tools to shed light on whether testosterone has a causal effect on aggressive behavior in the American Tree Sparrow. Write your results to be read in paragraph form, but put the letters (a), (b), (c), (d), (e), (f), (g), (h), (i) to delineate where you have included the following information. Include your code in your solutions.

   1. the estimated error standard deviation

   2. quantitative statistical evidence against the null (statistic / p-value)

   3. all the assumptions of the null model you are using

   4. quantitative statistical evidence concerning whether scientifically meaningful differences appear

      consistent with the data

   5. the scope of statistical inference (both population and causal)

   6. the scope of statistical interest (both population and causal)

   7. what assumptions would need to be met in order to make conclusions concerning the scope of

      interest

   8. your overall conclusions concerning the research question

   9. clearly denote what one improvement would likely help to reduce the variability of the data, and consequently increase the power of testing this causal effect

   10. data base:

       "cont" "exp"

       1. 3 4

       2. 4 5

       3. NA 3

       4. 3 6

       5. 3 2

       6. 4 4

       "7" 4 4

       "8" 3 3

       "9" 4 4

2. Assume that you are interested in conducting almost the same study as in question 2, but now with Rock Sparrows. The only other differences between the studies would be that (for reasons not discussed here, but related to a parallel experiment) you would like to have only 1/3 of the total birds in the study receive the testosterone treatment. Conduct a sample size calculation for estimating an average difference of 0.75 in aggression score with a power of 0.90 at an alpha level of 0.05. Assume that the Rock Sparrow is very similar to the American Tree Sparrow. Use any relevant information from question 2 to help with your sample size calculation. Include both your code and result in your solutions.