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data from one variable
grouped and analyzed data from two variables of a population
Contingency table or two-way table
- a frequency distribution for bivarite data.
- Cells: small boxes inside the rectangleformed by the heavy lines
- row - horizontal
- column - vertical
- Colculating total number: by summing the row totals, or column totals or the frequencies in the 20 cells of cont. table.
- they correspond to events represented in the margin of the contingency table
- Total of the column or a row devided by the total of all columns or rows
- ex. P(A1) = f/N = 381/1164=0.327
- probabilities for joint events
- P(A1&R2) = f/N = 3/1164 = 0.003 in the cells
- Joint probability distribution: joint probabilities are displayed instead of frequencies
- The row and column labels "total" add up to 1.000 and replaced to P(Ai) and P(Ri)
- P(B/A) "the probability of event of B given A"
- A is the given eventP(B/A)=P(A&B)
General multiplication rule
P(A&B)= P(A) *P(B/A) writinga tree diagrom is helpful when applying general multiplication rule
- a formula for computing joint probabilities in terms of mariganl and conditional probabilities.
- When the joint and marginal probabilities are known or can be easily determined directly, we use the conditonal probability rule to obtain conditional probabilities.
- When marginal and conditional probabilities are known or can be easily determined directly, we use the general multiplication rule to obtain joint probabilities.
- Event B is independent of event A if P(B/A) = P(B)
- If two events are not independent then they are dependent events
Special multiplication rule (for 2 independent events)
- IF A and B are independent events, then
- P(A & B) = P(A)* (P(B)
- IF P(A & B) equal to P(A)* (P(B), then A and B are independent events
Special multiplication rule (for 3 or more indep. events)
- If events A, B, C,.... are independent, then
- P(A&B&C&...)= P(A) * P(B) * P(C)
The Basic Counting Rule (BCR)
- r - different actions
- Suppose that r actions are to be performed in a definite order. Further suppose that there are m1 possibilities for the first action and that corresponding to each of these possibilites are m2 possibilities for the second action, and so on. Then there are m1*m2....mr possibilities altogether for the r actions.
- ex. on pg 214
- number of r=2 selectinga a model and elevation
- there are 4 possibilities for model , m1=4
- 3 posibilities for elevation, m2=3
- k - is a positive integer(counting number
- k factorial denoted as k!
- The factorial of a counting number is obrtained by successeively multiplying it by the next smaller counting number until reaching 1.
- k! = k(k-1)....2times 1
- 0! =1
- ex. 4!=4*3*2*1 = 24
The Permutations Rule
- The number of possible permutations of r objects from a collection of m objects is given by the formula
- mPr= m!
- the order metter
- AB and BA are different
- The number of possible permutations of r objects that can be formed from a collection of m ojbects is denoted mPr
Special permutation rule
- Finding the number of possible permutations among themselves.
- mPm = m!/(m-m)!= m!/0!=m!/1=m!
- the number of possible permutations of m ojbects among themselves is m!
The Combinations rule
- A combination of r objects froma collection of m objects is any unoreded arrangement of r of the m ojbects - any subset of r objects from the collection of m objects, denoted mCr (m choose r)
- AB and BA are the same - list only once
- mCr = m!