Sixty People Were Contacted And Responded To A Movie Survey
Sixty People Were Contacted And Respond To A Movie Survey The Followi
Sixty people were contacted and responded to a movie survey. The following information was obtained: (a) 6 people liked comedies, dramas, and science fiction; (b) 13 people liked comedies and dramas; (c) 10 people liked comedies and science fiction; (d) 11 people liked dramas and science fiction; (e) 26 people liked comedies; (f) 21 people liked dramas; (g) 25 people liked science fiction. Use a Venn diagram to illustrate the survey's results. Show how you arrived at your answer.
Paper For Above instruction
To analyze the survey data effectively, we will employ a Venn diagram approach to visualize the overlapping preferences among the three genres—comedy (C), drama (D), and science fiction (SF). The objective is to determine the number of respondents who liked exactly one, two, or all three genres, based on the given data.
Step 1: Defining Variables
Let us define the variables for each segment of the Venn diagram:
- \( a \): Number of people who liked only comedies (C), but not D or SF.
- \( b \): Number of people who liked only dramas (D), but not C or SF.
- \( c \): Number of people who liked only science fiction (SF), but not C or D.
- \( x \): Number of people who liked exactly two genres, specifically:
- \( x_1 \): Comedy and Drama only (\( C \cap D \) excluding \( C \cap D \cap SF \))
- \( x_2 \): Comedy and Science Fiction only (\( C \cap SF \) excluding \( C \cap D \cap SF \))
- \( x_3 \): Drama and Science Fiction only (\( D \cap SF \) excluding \( C \cap D \cap SF \))
- \( y \): Number of people who liked all three genres (\( C \cap D \cap SF \)).
From the question:
- \( y = 6 \) (liked all three).
- \( x_1 + y = 13 \) (comedy and drama), so \( x_1 = 13 - 6 = 7 \).
- \( x_2 + y = 10 \) (comedy and science fiction), so \( x_2 = 10 - 6 = 4 \).
- \( x_3 + y = 11 \) (drama and science fiction), so \( x_3 = 11 - 6 = 5 \).
Total liking comedy:
\[ |C| = a + x_1 + x_2 + y = 26 \]
Total liking drama:
\[ |D| = b + x_1 + x_3 + y = 21 \]
Total liking science fiction:
\[ |SF| = c + x_2 + x_3 + y = 25 \]
Total respondents:
\[ a + b + c + x_1 + x_2 + x_3 + y = 60 \]
Now, we substitute known values:
- \( x_1 = 7 \)
- \( x_2 = 4 \)
- \( x_3 = 5 \)
- \( y = 6 \)
Using the sums for each genre:
For \( |C| = 26 \):
\[ a + 7 + 4 + 6 = 26 \Rightarrow a = 26 - 17 = 9 \]
For \( |D| = 21 \):
\[ b + 7 + 5 + 6 = 21 \Rightarrow b = 21 - 18 = 3 \]
For \( |SF| = 25 \):
\[ c + 4 + 5 + 6 = 25 \Rightarrow c = 25 - 15 = 10 \]
Finally, to find the total, sum all segments:
\[ a + b + c + x_1 + x_2 + x_3 + y \]
\[= 9 + 3 + 10 + 7 + 4 + 5 + 6 = 44 \]
Since the total survey size is 60, and the counts sum to 44, there are:
\[ 60 - 44 = 16 \]
respondents who did not like any of these genres.
Summary of Results:
- Only comedy: \( a = 9 \)
- Only drama: \( b = 3 \)
- Only science fiction: \( c = 10 \)
- Comedy and Drama only: \( x_1 = 7 \)
- Comedy and Science Fiction only: \( x_2 = 4 \)
- Drama and Science Fiction only: \( x_3 = 5 \)
- All three genres: \( y = 6 \)
- None of these genres: 16
Constructing the Venn Diagram:
The Venn diagram will show three circles overlapping, with the counts placed in respective regions:
- Inside the Comedy circle only: 9
- Inside the Drama circle only: 3
- Inside the Science Fiction circle only: 10
- Intersection of Comedy and Drama only (excluding the center): 7
- Intersection of Comedy and Science Fiction only (excluding the center): 4
- Intersection of Drama and Science Fiction only (excluding the center): 5
- Intersection of all three:
Reasoning through the diagram ensures clear understanding of the distribution of preferences among respondents. This approach provides a visualization of how respondents' preferences overlap across the three genres, illustrating complex preferences succinctly.
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