An incomplete contingency table is provided. Use this table to complete the following.
a. Fill in the missing entries in the contingency table.
b. Determine [tex]P\left(C_1\right), P\left(R_2\right)[/tex], and [tex]P\left(C_1 \& R_2\right)[/tex].
c. Construct the corresponding joint probability distribution.
Complete the joint probability distribution.
Answer (1)
Complete the contingency table by filling in the missing values using subtraction and addition.
Calculate the probabilities P ( C 1 ) , P ( R 2 ) , and P ( C 1 & R 2 ) by dividing the corresponding totals and intersection by the grand total.
Construct the joint probability distribution by dividing each cell in the contingency table by the grand total.
The probabilities are: P ( C 1 ) = 0.5 , P ( R 2 ) = 0.6 , P ( C 1 & R 2 ) = 0.4 , and the joint probability distribution is constructed accordingly. P ( C 1 ) = 0.5 , P ( R 2 ) = 0.6 , P ( C 1 & R 2 ) = 0.4
Explanation
Problem Analysis We are given an incomplete contingency table and asked to complete it, calculate some probabilities, and construct the corresponding joint probability distribution.
Complete Contingency Table First, let's complete the contingency table. We can find the missing values using the information provided.
We know that the total for row R 1 is 16, and C 1 in R 1 is 4. Therefore, C 2 in R 1 is 16 − 4 = 12 .
We know that the grand total is 40, and the total for row R 1 is 16. Therefore, the total for row R 2 is 40 − 16 = 24 .
We know that C 2 in R 2 is 8, and the total for row R 2 is 24. Therefore, C 1 in R 2 is 24 − 8 = 16 .
We can now find the column totals. The total for column C 1 is 4 + 16 = 20 .
The total for column C 2 is 12 + 8 = 20 .
The completed contingency table is:
C 1
C 2
Total
R 1
4
12
16
R 2
16
8
24
Total
20
20
40
Calculate Probabilities Next, we need to determine P ( C 1 ) , P ( R 2 ) , and P ( C 1 & R 2 ) .
P ( C 1 ) is the probability of C 1 , which is the total of C 1 divided by the grand total: P ( C 1 ) = 40 20 = 0.5 .
P ( R 2 ) is the probability of R 2 , which is the total of R 2 divided by the grand total: P ( R 2 ) = 40 24 = 0.6 .
P ( C 1 & R 2 ) is the probability of both C 1 and R 2 occurring, which is the value at the intersection of C 1 and R 2 divided by the grand total: P ( C 1 & R 2 ) = 40 16 = 0.4 .
Construct Joint Probability Distribution Finally, we need to construct the joint probability distribution. This is done by dividing each cell in the contingency table by the grand total.
C 1
C 2
Total
R 1
40 4 = 0.1
40 12 = 0.3
40 16 = 0.4
R 2
40 16 = 0.4
40 8 = 0.2
40 24 = 0.6
Total
40 20 = 0.5
40 20 = 0.5
40 40 = 1.0
Final Answer Therefore: a. The completed contingency table is:
C 1
C 2
Total
R 1
4
12
16
R 2
16
8
24
Total
20
20
40
b. P ( C 1 ) = 0.5 , P ( R 2 ) = 0.6 , and P ( C 1 & R 2 ) = 0.4 .
c. The completed joint probability distribution is:
C 1
C 2
Total
R 1
0.1
0.3
0.4
R 2
0.4
0.2
0.6
Total
0.5
0.5
1.0
Examples
Contingency tables and joint probability distributions are used in market research to analyze the relationship between different variables, such as customer demographics and product preferences. For example, a company might use a contingency table to determine the probability that a customer in a certain age group prefers a particular product. This information can then be used to target marketing campaigns more effectively. Understanding these probabilities helps businesses make informed decisions about their marketing strategies and product development.
