Texts: A marketing team is going to test a product in one randomly selected area. Each of the possible market areas can be designated as either Urban (U) or Rural (R). In addition, some of the market areas are categorized as East (E) and others as West (W). There are a total of 125 possible market areas to choose from, 75 of which are East - the others are West. Also, out of the 125 possible market areas, 45 are Urban and the others are Rural. Finally, it is known that 25 of the 125 possible areas are both Urban and in the East.
a) Create a 2 by 2 contingency table with Location being the row variable and "Population Density" being the column variable. In this 2 by 2 table, include ONLY the information provided in the paragraph above. This will leave part of the table blank.
| | Urban | Rural |
|----------|-------|-------|
| East | | |
| West | | |
b) Now, by using the row and column totals that were given, deduce and fill in the remaining entries in the contingency table so that no blank cells remain. Do not fill in the table from part a). Create a new fully completed table for part b).
| | Urban | Rural |
|----------|-------|-------|
| East | 25 | 50 |
| West | 20 | 30 |
c) Using the row and column totals as well as the fact that there are 125 total possible market areas, what are P(E), P(W, U), and P(R) if we randomly pick one site?
P(E) = 75/125 = 3/5
P(W, U) = 20/125 = 4/25
P(R) = 80/125 = 16/25
d) Interpret the meaning of P(W, R) and find its probability.
P(W, R) represents the probability of selecting a market area that is both Rural and in the West. Its probability can be calculated as follows:
P(W, R) = 1 - P(E, R) - P(W, U) = 1 - (50/125) - (4/25) = 1 - 2/5 - 4/25 = 1 - 10/25 - 4/25 = 1 - 14/25 = 11/25
e) Interpret the meaning of P(W, U, R) and find its probability.
P(W, U, R) represents the probability of selecting a market area that is both Urban, Rural, and in the West. Since it is not possible for a market area to be both Urban and Rural at the same time, P(W, U, R) = 0.
f) True or False: P(R) = 1 - P(U). Why or why not?
False. P(R) does not equal 1 - P(U) because there is a possibility of market areas being both Urban and Rural. Therefore, P(R) = 80/125 ≠1 - P(U).
g) Find the probability of the selected site being urban if we know that it has been selected to be in the East.
P(U|E) = P(U, E) / P(E) = 25/125 / 75/125 = 25/75 = 1/3
h) Find the probability of the selected site being in the East if we know that we have selected a site that is urban. Is your answer the same as in part g)?
P(E|U) = P(E, U) / P(U) = 25/125 / 45/125 = 25/45 ≠1/3. The answer is not the same as in part g.
i) Referring back to part d, is P(W, R) = P(W) * P(R)? What does your answer imply about location and population density?
No, P(W, R) ≠P(W) * P(R). This implies that the events of being in the West and being Rural are not independent of each other. The probability of both events occurring together is not equal to the product of their individual probabilities.
By cross-multiplying in this formula, we get that P(W, U) = P(W) * P(U). This is called the multiplication rule. Insert the appropriate three numbers into this multiplication rule and show that it is true for our experiment.
P(W, U) = (20/125) = (20/125) * (45/125) = P(W) * P(U)
