a local television station sends out questionnaires to determine if viewers would rather see a documentary, an interview show or reruns of a game show there were 350 responses with following results. 105 were interested in an interview show and a documentary but not reruns.14 were interested in an interview show and reruns but not a documentary.49 were interested in reruns but not an interview show. 84 were interested in a interview show but not a documentary. 35 were interested in a documentary and reruns.21 were interested in an interview show and reruns. 28 were interested in none of the above
148
To determine how many viewers would prefer each type of show, we can use a method called the Principle of Inclusion-Exclusion. Let's break down the information given:
Total responses: 350
Interested in interview show and documentary but not reruns: 105
Interested in interview show and reruns but not a documentary: 14
Interested in reruns but not an interview show: 49
Interested in interview show but not a documentary: 84
Interested in documentary and reruns: 35
Interested in interview show and reruns: 21
Interested in none of the above: 28
To find the number of viewers interested in each individual type of show, we can follow these steps:
1. Start with the total number of responses (350).
2. Subtract the number of viewers interested in none of the above (28).
3. Subtract the number of viewers interested in both an interview show and documentary but not reruns (105).
4. Subtract the number of viewers interested in both an interview show and reruns but not a documentary (14).
5. Subtract the number of viewers interested in both a documentary and reruns but not an interview show (35).
Let's calculate:
Total responses (step 1): 350
Minus viewers interested in none of the above (step 2): 350 - 28 = 322
Minus viewers interested in both an interview show and documentary but not reruns (step 3): 322 - 105 = 217
Minus viewers interested in both an interview show and reruns but not a documentary (step 4): 217 - 14 = 203
Minus viewers interested in both a documentary and reruns but not an interview show (step 5): 203 - 35 = 168
After following these steps, we find the following totals:
Number of viewers interested in an interview show: 168
Number of viewers interested in a documentary: 217
Number of viewers interested in reruns: 203
Therefore, according to the survey, 168 viewers would prefer to see an interview show, 217 viewers would prefer to see a documentary, and 203 viewers would prefer to see reruns of a game show.
To analyze the data provided, we can use a Venn diagram to visualize the categories and their relationships. Let's break down the information step-by-step:
Step 1: Determine the number of respondents interested in each option individually.
- Interested in an interview show: 105 (given)
- Interested in a documentary: x (unknown)
- Interested in reruns: y (unknown)
Step 2: Identify the overlapping categories.
- Interested in an interview show and a documentary but not reruns: 105 (given)
- Interested in an interview show and reruns but not a documentary: 14 (given)
- Interested in reruns but not an interview show: 49 (given)
- Interested in an interview show but not a documentary: 84 (given)
- Interested in a documentary and reruns: 35 (given)
- Interested in an interview show and reruns: 21 (given)
- Interested in none of the above: 28 (given)
Step 3: Use the formula to find the number of respondents in each category.
The formula to calculate the number of respondents in each category is:
Total = A + B + C + D + E + F + G + H + I
Where:
A = Interested in an interview show and a documentary but not reruns (105)
B = Interested in an interview show and reruns but not a documentary (14)
C = Interested in reruns but not an interview show (49)
D = Interested in an interview show but not a documentary (84)
E = Interested in a documentary and reruns (35)
F = Interested in an interview show and reruns (21)
G = Interested in a documentary but not an interview show (x)
H = Interested in reruns but not a documentary or an interview show (y)
I = Interested in none of the above (28)
Now let's plug in the obtained values into the formula:
350 respondents = 105 + 14 + 49 + 84 + 35 + 21 + x + y + 28
Step 4: Simplify the equation.
350 = 316 + x + y
Step 5: Solve for x and y.
We know that x + y + 28 = 350 - 316 = 34
Since we don't have any further information about the amount of respondents interested only in a documentary (x) or only in reruns (y), we cannot determine their values explicitly.
Therefore, we can say that:
- The number of respondents interested only in an interview show is 84.
- The number of respondents interested in both a documentary and an interview show is 105.
- The number of respondents interested in both an interview show and reruns is 21.
- The number of respondents interested in both a documentary and reruns is 35.
- The number of respondents interested in reruns only is 49.
- The number of respondents interested in none of the above is 28.
Please note that we didn't determine the exact number of respondents interested only in a documentary (x) or only in reruns (y).