[2] (30 points) During early evening mountain goats come down to water in a Montana location at a rate of 5 per hour. Consider each goat's arrival independent of the next with arrivals consistent with the rate of 5 per hour. Assume a Poisson model applies. [a] If there is a 3-hour block of time considered "early evening" what is the probability that more than 10 goats will come down to water in those 3 hours? [b] Your instructor has just seen one goat but only has an hour more before he must go. What is the probability that he fails to see another goat before leaving? [c] A new camera system purports to estimate an animal's height and weight from the image processed through its system. The concern is that excessive drought is limiting the supply of grasses and lichens at higher altitudes and that weights may be reduced from that expected for a given height. Height estimates are considered very good, so expected weight (from previous literature, based on height) will also be considered good. Camera-processed weight reliability is at question. Assume that weight difference from expected is distributed normally. Fifteen goats measured found that camera determined weight less expected weight had a mean and sample standard deviation of x= 8.2 lb, s = 3.1 lb. Test the null hypothesis H0: = 0 vs : Ha< 0 using the appropriate test. Use = 0.05. [d] Comment on the practical implications of and define making a Type I error from a wildlife management perspective and define and discuss the implications of making a Type II error for the test in [c] from that same perspective. [e] Someone challenged the Poisson model for goat arrival to water. Over 90 evenings during the 3-hr early evening block, the recorded number of evenings in which 0 to 10 goats were observed was only 5; 11 to 15 goats were observed 50 times; 16 to 20 goats were observed 20 times; and greater than 20 goats were observed 15 times. Assume the rate is 5/hr, as noted above, and observe goats in the 4 intervals. Find the probability for 0-10, 11-15, 16-20, and 20- under the null hypothesis. Then use the Chi-Square goodness of fit test to test the Poisson model assumption. Use alpha = 0.05. [f] List the p-value for this test.