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]$P(C_1)$[/tex], [tex]$P(R_2)$[/tex], and [tex]$P(C_1 \& R_2)$[/tex].
c. Construct the corresponding joint probability distribution.
| | [tex]$C_1$[/tex] | [tex]$C_2$[/tex] | Total |
| :---- | :----------- | :----------- | :---- |
| [tex]$R_1$[/tex] | 4 | | 16 |
| [tex]$R_2$[/tex] | | 8 | |
| Total | | | 40 |
a. Complete the contingency table.
| | [tex]$C_1$[/tex] | [tex]$C_2$[/tex] | Total |
| :---- | :----------- | :----------- | :---- |
| [tex]$R_1$[/tex] | 4 | | 16 |
| [tex]$R_2$[/tex] | | 8 | |
| Total | | | 40 |
(Type whole numbers.)
b. Find each probability.
[tex]$P(C_1)=[/tex] (Type an integer or decimal rounded to two decimal places as needed)
Answer (1)
Complete the contingency table by subtracting known values from totals to find missing entries.
Calculate the probabilities P ( C 1 ) , P ( R 2 ) , and P ( C 1 & R 2 ) by dividing the corresponding values by the total number of observations.
Construct the joint probability distribution by dividing each cell in the contingency table by the total number of observations.
The probabilities are: P ( C 1 ) = 0.4 , P ( R 2 ) = 0.4 , P ( C 1 & R 2 ) = 0.3 , and the completed contingency table allows us to derive the joint probability distribution.
Explanation
Analyze the problem We are given an incomplete contingency table and asked to complete it, find some probabilities, and construct the joint probability distribution. Let's start by completing the table.
State given values The contingency table has rows R 1 and R 2 , and columns C 1 and C 2 . The total number of observations is 40. We are given the following values: R 1 C 1 = 4 , R 1 T o t a l = 16 , R 2 C 2 = 8 , T o t a l = 40 .
Calculate missing values We can find the missing values using the given information. R 1 C 2 = R 1 T o t a l − R 1 C 1 = 16 − 4 = 12 .
C 2 T o t a l = T o t a l − R 1 T o t a l = 40 − 16 = 24 .
R 2 T o t a l = T o t a l − R 1 T o t a l = 40 − 16 = 24 . However, R 2 T o t a l = R 2 C 1 + R 2 C 2 , so R 2 T o t a l = C 2 T o t a l − R 2 C 2 = 24 − 8 = 16 .
C 1 T o t a l = T o t a l − C 2 T o t a l = 40 − 24 = 16 .
R 2 C 1 = C 1 T o t a l − R 1 C 1 = 16 − 4 = 12 .
Completed contingency table So, the completed contingency table is:
C 1
C 2
Total
R 1
4
12
16
R 2
12
8
16
Total
16
20
40
Calculate probabilities Now, let's calculate the probabilities: P ( C 1 ) = T o t a l C 1 T o t a l = 40 16 = 0.4 .
P ( R 2 ) = T o t a l R 2 T o t a l = 40 16 = 0.4 .
P ( C 1 & R 2 ) = T o t a l R 2 C 1 = 40 12 = 0.3 .
Construct joint probability distribution Finally, let's construct the joint probability distribution table by dividing each cell in the contingency table by the total number of observations (40):
C 1
C 2
R 1
40 4 = 0.1
40 12 = 0.3
R 2
40 12 = 0.3
40 8 = 0.2
State final answer Therefore, a. The completed contingency table is:
C 1
C 2
Total
R 1
4
12
16
R 2
12
8
16
Total
16
24
40
b. P ( C 1 ) = 0.4 , P ( R 2 ) = 0.4 , and P ( C 1 & R 2 ) = 0.3 .
c. The joint probability distribution is:
C 1
C 2
R 1
0.1
0.3
R 2
0.3
0.2
Examples
Contingency tables and probability distributions are used in market research to analyze customer preferences. For example, a company might survey customers to determine their favorite product features (columns) based on their age group (rows). The probabilities calculated from the table can then be used to target specific customer segments with tailored marketing campaigns, maximizing the effectiveness of their advertising budget. Understanding these relationships helps businesses make informed decisions about product development and marketing strategies.
