Second Homework Assignment for ENVS 355
Covering the Z-test, The KS-test, and Poisson Statistics

Each data set has 36 values in it. The data sets represent monthly rainfall in Eugene from 1973-75, 1983-85, and 1993-95.

      Data Set 1:


        J   F   M     A  M   J   J   A   S   O   N   D

      12.8 8.4 12.5 2.5 1.1 0.4 1.4 0.4 0.1 1.6 6.4 9.3
      6.9 6.8 7.6 2.9 2.2 0.9 1.2 2.1 0.1 5.7 8.5 7.1
      9.8 7.6  6.2 1.9 0.9 0.2 0.4 2.0 1.1 1.9 1.3 1.2

      Data Set 2:

      6.8 12.3 10.6 3.4 1.8 1.8 1.8 3.2 0.5 1.4 13.1 7.5
      2.1 9.6 6.4 5.4 3.9 3.9 0.3 0.1 0.9 6.1 18.7 4.6
      0.3 5.1 5.7 0.5 1.5 2.5 1.4 0.1 2.1 4.8 6.3 3.5

      Data Set 3:

      6.9 2.6 8.6 7.9 6.9 3.7 1.1 1.8 0.1 1.5 2.0 10.8
      5.5 5.5 5.5 2.0 1.6 1.1 0.1 0.1 2.1 7.5 9.6 6.1
      15.4 3.8 6.4 5.6 2.1 2.3 1.1 1.0 1.1 3.9 9.5 13.4

1. Using the Z-test, determine if there is any significant difference in the sum of the January through April rainfall within these three time periods.

2. Now using the KS test compute the D_max statistic between the 3 data sets (using all twelve months) and determine from that statistic if any of the data sets are significantly different from one another.

3. Compare the data sets to model Gaussian(Normal) distributions using the procedure we went through in class with Excel Random number Generator:

• In the first case compare each data set to a model guassian (Normal) distribution with mean = 4.5 and standard deviation = 4.0. and compute D_max. You should find, in all cases, this model to be a poor fit to the data. If it is, what can you conclude about the nature of the data? What aspect of the model seems to not make any physical sense in comparison to the data?

• Now combine all three data sets into one average data set and find the best fitting Normal Distribution to that data. In this case, comparing 36 points against a model, Dmax should be less than 0.23 for the model to be acceptable.

Read each question carefully. In the context of the question the important thang that you need to determine is the average rate of occurence.

Important math note:

0! = 1 (by definition)

Questions:

1. Army records noted the chance of a Prussian cavalryman being killed by the kick of a horse. The records of ten army corps were examined over 20 years, an a total of 122 deaths by horse kick were noted. What is the probability that 3 cavalyrman would be killed in a single year?

2. It has been observed that the average number of reportable traffic accidents on the Hollywood Freeway between 7 and 8 AM on Wednesday mornings is 1 per hour. What is the chance that there will be 2 accidents on the Freeway some given Wednesday morning during this time interval?

3. It is 8 PM in Nebraska, near the end of summer, and, our heroine, Sunshine Moonbeam is watching fireflies on the back porch. From the last few nights on the porch at that hour, she knows that the average number of firefly flashes is 6 per minute. She refuses to go to bed until 2 minutes pass in which she observes no flashes - that way she knows that the fireflies have said good night to her. What is the probability of observing 0 flashes in a 2 minute period?

4. The switchboard in a small Denver law office gets an average of 2.5 incoming phonecalls during the noon hour on Thursdays. Experience shows that the reduced staffing during lunch can handle up to 5 calls in that hour. What is the chance of 6 calls being received during that period, some particular Thursday?

5. Security cameras in the Bullion Bank of Nevada have established that, during the last half-hour of business on Friday, on average 3 customers enter its lobby to deposit their bulk gold. Gold is heavy, and the bank does not want to keep its gold-bearing customers waiting. How many tellers should it have on duty, at that time, so that there is only a 10% chance or less that any customer will be kept waiting for a teller?

6. 6. Idaho's Trout'R'Us lake is continually stocked with trout. Anglers average catching 2 trout per hour. Dweebo's girlfriend, Dweizel is a dedicated fisherman. Dweebo know's he will win Dweizel's hand if she catches a trout while fishing together. Since Dweizel can not stand incompetence, she can not tolerate catching 0 fish. Plus, Dweizel is really impatient and can only stand to fish (with Dweebo) for 0.5 hours. What is the probability that Dweebo will not win her hand (or anything else ...)?

7. Over the last 500 million years there have been 31 asteroid impacts that have extincted at least 1/3 of all life on the planet. What is the probability that the next extinction event will occur during the next 1 million years?

8. Over the last 110 years, there have been 7 Category 5 Storms that have have passed through a latitude longtitude grid cell defined by Latitude = 27-28 degrees and Longtitude defined by 85-86 degrees. In the year 2005, two Cat 5 Storms passed through this grid cell. What is the probability of that occurring.