Geography 360

Problem Set Four: Working with Probability Distributions

1) The Wisconsin Department of Natural Resources collects data on the sightings of rare animals in the state. Follow the link to reported Pine Martin sightings. On pages 132 and 133 you will find a table listing sightings by county for a number of mammals. If we assume that reported sightings were a good estimate of Pine Martin populations in each county [Why might this be a shaky assumption?], how might we determine if there is a random distribution of Pine Martin in the counties listed. That is, which probability distribution would best simulate the expected number of sightings in each county? How does the reported data compare to the expected data? Can you provide justification for your results?

2) State Street is home to numerous restaurants. Would you expect these restaurants to be randomly distributed along the street? A map of the Madison Central Business District (including State Street) is available at the campus information center and at several merchants along State Street. Taking a copy of this map, divide State Street into twelve zones, either north or south of State Street and east or west of the Lake, Frances, Gilman, Gorham, Johnson, Dayton, and Mifflin Street intersections. Hold would you provide statistical evidence of the relative randomness of restaurant locations on State Street. Do you find the results surprising? Why or why not?

3) A) Geographers have been very active in the analysis of the diffusion of innovations across space. In the most basic form, analyses suggest that each person that is introduced to an innovation (for example each farmer introduced to new cultivation methods) has a given probability of adopting. This probability is usually taken from early introductions of the innovation in small areas. If we found that 1 out of 3 farmers adopted a new cultivation technique upon being introduced to it, how might we calculate the expected number of adopters for an extension agent during one month (four weeks) if he meets with an average of 10 farmers per week. Draw a distribution histogram for the probability of getting 0, 1, 2, …. 40 adopters during the month.

B) Using a die (to provide a probability of 1/3), design a method of modeling the response of 40 farmers. How does your modeled result compare to the expected random result calculated above?

4) A) While social characteristics of human populations are usually not randomly distributed, physical characteristics are often considered to be so. If we take the following numbers are a random sample of heights (in cm) for five year old boys (attention: height would be a continuous variable, right?), how could we calculate the probability of a particular boy being at, above, or below a certain height? (Hint: Your first step would be to assume that heights are normally distributed – although there are some very small boys and some very tall boys, most are near the same height. Second step: calculate the mean and standard deviation for the sample and standardize the respective deviations to allow comparison with the Normal Table in your text.)

105.9 110.0 106.7 108.5 107.7 114.3 108.6 104.6 113.7

116.7 103.5 96.1 110.8 97.2 109.6 110.5 105.9 106.2

B) What is the probability of a five-year-old boy being taller than 108 cm?

C) What is the probability of being smaller than 100 cm?

D) What is the probability of being between 98 and 103 cm?

E) If another sample of 30 boys were taken, what height would you expect 28 of the boys to be smaller than?

F) What would be the smallest expected height from a sample of 100 five-year-old boys?

Below, I have provided examples of Excel entries to help with calculations. In order to use the examples, type the characters as entries in the respective cells of an Excel spreadsheet. (The A, B, C and left column of numbers refer to the column and row designations in the spreadsheet.) Words will appear as written. Formulas will leave blank cells or error messages until you provide the necessary values for events and outcomes in the second row of each example. Where I have typed etc., you should follow the pattern in the column the necessary number of times. You can also copy (control C) a formula once entered and paste it into the necessary number of rows.

Binomial Distribution

A B C

1	Number of Events	Probability of desired outcome	Probability of all other outcomes
2			=1-B2
3	Frequency	probability of frequency	expected events of given frequency
4	0	=((FACT(A$2))*(B$2^A4)*(C$2^(A$2-A4)))/((FACT(A4)*(FACT(A$2-A4))	=A$2*B4
5	1	=((FACT(A$2))*(B$2^A5)*(C$2^(A$2-A5)))/((FACT(A5)*(FACT(A$2-A5))	=A$2*B5
6	2	etc.	etc.
7	3
8	4
	etc.

Poisson Distribution

A B C

1	number of events	frequency of outcomes	average frequency
2			=B2/A2
3	Frequency	probability of frequency	expected events of given frequency
4	0	=(C$2^A4)/((EXP(C$2))*(FACT(A4)))	=A$2*B4
5	1	=(C$2^A5)/((EXP(C$2))*(FACT(A5)))	=A$2*B5
6	2	etc.	etc.
7	3
8	4
	etc.