In A Recent Survey Of 100 Women The Following Information Wa

In A Recent Survey Of 100 Women The Following Information Was Gathere

In a recent survey of 100 women, the following information was gathered. 42 use shampoo A. 54 use shampoo B. 30 use shampoo C. 12 use shampoos A and B. 20 use shampoos A and C. 5 use shampoos B and C. 4 use all three. Use the figure to answer the question in the problem. How many are using shampoo A only (Region I)?

The problem involves understanding set intersections within a Venn diagram. We are given the total number of women and the number of women using each shampoo, as well as those using combinations of shampoos A, B, and C. To find the number of women who use only shampoo A (Region I), we can apply the principle of inclusion-exclusion.

Number of women using shampoo A only (Region I) is calculated as follows:

Start with total women using shampoo A: 42.

Subtract women who use A and B (12) and women who use A and C (20), then add back women who use all three (since they were subtracted twice): 4.

Thus: 42 - 12 - 20 + 4 = 14.

Therefore, 14 women use only shampoo A (Region I).

Paper For Above instruction

The detailed analysis of set intersections via Venn diagrams is fundamental in understanding overlapping populations in surveys and studies. In the context of the recent survey involving 100 women and their shampoo preferences, the primary goal was to determine how many women exclusively used shampoo A, represented as Region I in the Venn diagram.

Given data points include the total number of women who use each shampoo and their combinations: 42 women use shampoo A, 54 use shampoo B, and 30 use shampoo C. Additional information includes 12 women who use both A and B, 20 who use both A and C, 5 who use both B and C, and 4 who use all three shampoos.

To address this problem, we adopt the principle of inclusion-exclusion, a key combinatorial method used to avoid double-counting overlapping elements across different sets. The classical inclusion-exclusion principle for three sets states that the number of elements in the union of the three sets is: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|.

Calculating the number of women who use only shampoo A involves subtracting those who use A along with other shampoos from the total number who use A. Specifically, women who use A with B (12) and with C (20) are subtracted from the total who use A, but since women who use all three shampoos are included in both these overlaps, they are added back once to prevent double subtraction.

The formula derived and applied yields: 42 - 12 - 20 + 4 = 14. Thus, 14 women exclusively use shampoo A, corresponding to Region I in the Venn diagram.

This approach illustrates how set theory and combinatorics can be employed effectively to interpret survey data involving overlapping categories. Proper application of inclusion-exclusion guarantees accurate counts in complex overlapping data sets, a critical skill in statistical analysis, market research, and epidemiological studies.

Understanding such concepts allows researchers and analysts to interpret overlapping data correctly, enabling better decision-making and resource allocation based on the precise distribution of characteristics within populations.

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